1. INTRODUCTION

The vast majority of optical systems are optimized for wavelength independent focusing to provide high image quality, such as achromatic or apochromatic designs. The exact opposite goal, a significantly pronounced axial wavelength spreading (e.g., in the focus range), is pursued with hyperchromats, which are particularly interesting for applications such as surface profiling [1] or confocal microscopy [2,3]. A key measure describing the effectiveness of these optical systems is the equivalent Abbe number ${V_A}$, which should be minimized to achieve substantial axial chromatic spreading. This parameter is equal to the Abbe number that a solitary lens would need to attain the same axial chromatic spreading as the regarded optical system [4]. An optical system with an equivalent Abbe number that is lower than the Abbe number of the lens materials used is called a “hyperchromatic” system. The Abbe number is determined by the total focal length $f$ of the system and the longitudinal chromatic focus shift $\Delta {a^\prime}$ across the entire wavelength range, as in [5]

Several previous studies have focused on the optimum design of hyperchromats with the central aim of achieving the lowest possible equivalent Abbe number. In their work, Novák and A. Mikš [6] presented various triplets that achieve significant axial chromatic spreading. Therefore, they used a paraxial-based mathematical approach to determine the optimum optical power for each individual lens in the triplet, while ensuring a linear dependence of the system focal length on wavelength. However, the equation-based solutions obtained were not verified using optical design software, preventing an analysis of imaging properties and aberrations, which limits the usability for the design of real optical systems.

In [7], Yang et al. described the design of a highly dispersive lens group. They used a novel design methodology that promotes optimization through a single configuration instead of the typical three for each reference wavelength. The resulting optical system exhibited diffraction-limited imaging quality across the wavelength spectrum, with a linear relationship between the axial chromatic shift and wavelength. However, this system, consisting of seven optical elements, has achieved a modest axial chromatic shift of 0.38 mm in the wavelength range from 580 nm to 780 nm.

In earlier studies, we analyzed the influence of focal lengths and lens materials on hyperchromatic two-lens systems, first considering simple lens doublets with zero distance [8] and later introducing or allowing an additional air gap between the lenses [9]. Here, both pure refractive and also hybrid systems in the combination of diffractive and refractive components are taken into account. These studies provide important design principles for dual-lens hyperchromats and demonstrate extremely low equivalent Abbe numbers that apply to both refractive and hybrid systems. However, these results also raise the question of systems with an even smaller equivalent Abbe number. Therefore, in this work we investigate the possibilities of maximizing axial color spreading by combining two previously considered hyperchromatic approaches in series. Hereby, each single hyperchromat consists of two optical components, including purely refractive solutions as well as hybrid diffractive–refractive combinations. For the investigation, it is also required that all hyperchromatic systems under consideration exhibit linear axial chromatic behavior. This requirement enables a direct comparison with previously published hyperchromatic systems and simplifies measurement processes. Following the Introduction, Section 2 introduces starting systems consisting of two double-lens hyperchromats in series. Each individual unit of the double hyperchromat is designed as a two-lens system, optimized for maximum axial chromatic dispersion according to previous publications.

For these systems, the lens properties, whether they are converging or diverging, and the materials involved are explicitly specified. In Sections 3 and 4, these lens configurations are transferred into a ray-tracing-based optical design software tool, where the systems are optimized to achieve a minimum equivalent Abbe number. This optimization takes into account aberration effects as well as geometric constraints. This procedure allows design rules to be derived for the selection of the most suitable double-lens combinations for realizable double hyperchromats, offering extreme axial chromatic spreading. Both purely refractive and hybrid combinations are considered with extreme (ZnSe [10]) and conventional materials (SF66 [11]). In addition, it is possible to easily adapt the system for varying specifications, e.g., when changing the focal length for the reference wavelength.

2. INITIAL SELECTION OF DOUBLE-HYPERCHROMATIC STARTING SYSTEMS

In [9], we found that careful selection of the appropriate focal lengths and materials is crucial for a two-lens hyperchromat, aiming to achieve minimal equivalent Abbe numbers. Hereby, for both refractive and hybrid hyperchromats, it is necessary to include one lens with positive optical power and a second with negative optical power. Moreover, in the case of refractive hyperchromats, the positive lens should be made of a material with the lowest possible Abbe number, whereas the negative lens requires a material with the highest Abbe number. The combination of two glasses with very different Abbe numbers also enables the fulfillment of the condition for linear behavior in chromatic axial spreading [12].

With reference to our previous investigations, ${{\rm CaF}_2}$ [13], which has the highest Abbe number among the considered lens materials, and ZnSe [10], with an extremely small Abbe number, prove to be the most suitable materials for the studies. Consequently, the lenses used in this study are made exclusively from these two materials. In the pure refractive systems, each setup includes two hyperchromats, each consisting of two lenses: one with a positive focal length and the other with a negative focal length. The positive lenses are made of ZnSe, while the negative lenses are made of ${{\rm CaF}_2}$. In this context, there are four different basic lens orders for purely refractive systems that appear to be suitable for achieving exceptionally low equivalent Abbe numbers. In a final consideration, a more common material was used instead of ZnSe due to potential processing complexities. This decision was made in order to demonstrate the potential of two cascaded hyperchromats for axial chromatic spreading with conventional materials. SF66 [11], known for its minimum Abbe number among widely used classical inorganic glasses, was selected for this purpose.

For hybrid systems, our study was restricted to configurations encompassing a combination of a hybrid hyperchromat and a pure refractive one. Due to potential efficiency issues associated with the use of two diffractive components, double hybrid combinations are excluded. The hybrid hyperchromat is always composed of a diffractive element and a refractive element exclusively using ZnSe. The diffractive structure is directly integrated into the flat surface of a fused silica window. The alternating signs in focal lengths of the lenses in this hyperchromat provide flexibility, allowing arrangements where the refractive lens can be positive and the diffractive lens negative, or the reverse. The pure refractive hyperchromat combines a positive lens made of ZnSe and negative one made of ${{\rm CaF}_2}$. This specific selection is based on our previous results [9], which have shown that such combinations lead to the most effective hyperchromatic systems. Consequently, the characteristic criterion of alternating signs in the optical power of both constituents is still satisfied. Our analysis involves a total of 16 different hybrid systems, each of which was carefully derived from these specifications.

3. RAY OPTICAL OPTIMIZATION OF REFRACTIVE DOUBLE HYPERCHROMATIC SYSTEMS

The initial configurations introduced in the previous section served as input for an optimization procedure based on ray tracing using a commercial optical design software (OpticStudio [14]). The aim is to derive systems which, besides a low equivalent Abbe number, have a high imaging quality characterized by small spot diagrams along the axis and a linear correlation between the focus shift and wavelength. The required linearity is important as it ensures a constant axial resolution of the system over the entire wavelength spectrum and thus enables simplified signal processing.

 

Fig. 1. Illustration depicting the schematic ray traces for three reference wavelengths (${\lambda _{\rm d}}$, ${\lambda _{\rm F}}$, ${\lambda _{\rm C}}$) as they pass through two sequentially connected hyperchromats characterized by variable parameters including lens radii (${{\rm r}_{11}}$, ${{\rm r}_{12}}$, ${{\rm r}_{21}}$, ${{\rm r}_{22}}$, ${{\rm r}_{31}}$, ${{\rm r}_{32}}$, ${{\rm r}_{41}}$, ${{\rm r}_{42}}$) and interlens distances (${{\rm D}_{12}}$, ${{\rm D}_{23}}$, ${{\rm D}_{34}}$). The parameters ${{\rm a}_{\rm F}}$, ${{\rm a}_{\rm d}}$, and ${{\rm a}_{\rm C}}$ indicate the vertex distances measured from the final lens surface to the respective focal planes corresponding to the reference wavelengths.

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The initial parameters for the components, such as the focal lengths of the individual lenses, were determined based on the design principles outlined in Section 2. In cases where positive optical power was required, the lens was designed as a biconvex lens with an initial focal length of 20 mm, while a biconcave structure was chosen for negative optical power with a focal length of ${-}{20}\;{\rm mm}$. Initially, all distances between the lenses were assumed to be 5 mm and the lens thicknesses were set to 2.5 mm. All systems, both purely refractive and hybrid configurations, use a collimated input source with an entrance pupil diameter of 9 mm, with the total axial length limited to 100 mm. The total length of the system is measured from the first vertex of the first lens to the focal point of the system with the greatest distance to the lens.

Following the definition of initial parameters, the variable design parameters were specified. These variables, which comprise the lens radii, distances between the lenses, and distances to the image planes, were automatically adjusted by the ray-tracing software tool to determine the optimum settings based on the optimization criteria. For illustration, Fig. 1 schematically presents an initial setup, indicating the relevant parameters for a refractive system. The values ${{\rm a}_{\rm F}}$, ${{\rm a}_{\rm d}}$, and ${{\rm a}_{\rm C}}$ represent the vertex distances measured from the last lens surface to the respective focal points for different wavelengths (${\lambda _{\rm F}} = {486.13}\;{\rm nm}$, ${\lambda _{\rm d}} = {587.56}\;{\rm nm}$, ${\lambda _{\rm F}} = {656.27}\;{\rm nm}$).

To accelerate the process and avoid undesirable configurations, several constraints were imposed before the optimization process started. These restrictions include the demand that the focal lengths of the lenses with either positive or negative optical power must not change their sign. Additionally, the total length of the systems, measured from the first optical surface to the focal point of the wavelength with the largest imaging distance, was limited to a maximum of 100 mm. To avoid overlapping lens surfaces and ensure manufacturability, a minimum lens thickness of 1 mm was specified. Furthermore, in order to guarantee an adequate image quality, a minimum value for the Strehl ratio of 0.8 has also been specified, which is equivalent to a diffraction limited system [15].

The requirement for a focus shift that is linearly dependent on the wavelength was addressed by introducing supporting points. Specifically, the entire wavelength range from 486 nm to 656 nm was divided into four segments, each with an identical wavelength spacing. It is required that the calculated chromatic-dependent focus shifts are identical for all wavelength segments. The chromatic-dependent focus shift was computed for each segment, with the condition that they all exhibit an equivalent value for their focus shift. The main objective was to minimize the equivalent Abbe number while satisfying the specified boundary conditions.

To evaluate and optimize the performance of the systems in the program the specifications and constraints were defined as a linear set of weighted target values, called the merit function [16]. The value of the merit function (${M_F}$) is calculated by the deviation of the actual value from the target value and the according weighting parameter (${W_i}$) for all specifications. Through an iterative variation of the variable parameters, the program tries to minimize the value of the merit function. If all current values (${V_i}$) approach their targets (${T_i}$), the merit function will be zero. Here, the indices $i$ describe the individual target parameters mentioned above. When the value of the merit function does not decrease with further iterations, the optimization stops. Mathematically the merit function MF is defined as [17]

Using the methodology outlined, the following results displayed in Table 1 are obtained for the four systems analyzed, listed with their focal lengths $f$, materials, interlens distances D, axial chromatic shift $\Delta {a^\prime}$, and equivalent Abbe number ${V_A}$.

Table 1. Parameters of Optimized Pure Refractive Systems^a

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As a result of Table 1, all four optimized configurations show very small equivalent Abbe numbers with the lowest value of 0.987 for Configuration 4. This value of Configuration 4 is notably eight times lower than the Abbe number of the highly dispersive material ZnSe and more than three times lower than the Abbe number of a single diffractive element. An interesting observation across all configurations is that the order of the lenses plays a crucial role. Thus, it proves to be more advantageous to alternate a highly dispersive positive lens with a low dispersive negative lens. Also, it is essential to consistently avoid the arrangement of two negative lenses consecutively, as exemplified in Configuration 2, which leads to the highest equivalent Abbe number in this comparison. For illustration, the optical models and ray traces of the optimized configurations are depicted in Fig. 2. Figures 2(a)–2(d) display the lens cross-sections and ray paths for Configurations 1 to 4. All optimized configurations achieved a regression coefficient of over 99% for the linear behavior of the wavelength-dependent focus position.

 

Fig. 2. Cross sections and ray traces for the four calculated refractive cascaded hyperchromatic configurations after optimization. Configurations 1 to 4 are assigned to the designations (a) to (d).

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The ray traces obtained from the optimized systems show a clear trend: Systems in which the rays travel the longest distances through the lens system have the lowest equivalent Abbe numbers [see Figs. 2(a) and 2(d)]. This is consistent with the use of alternating signs in the focal lengths in these systems. The positive highly dispersive lenses induce the spectral decomposition, an effect that is more pronounced at shorter focal lengths (first and third lens of Configuration 1; and second and fourth lens of Configuration 4). The only limiting factor on the focal length of the positive lenses is the introduction of spherical aberrations, which increase with smaller curvatures. The negative lenses in these systems, which are characterized by low dispersion, do not reduce the spectral decomposition. Instead, their task is to extend the light path through the lens system by steering the marginal rays to the edges of the following lenses and to compensate spherical aberrations. Consequently, Configurations 2 [Fig. 2(b)] and 3 [Fig. 2(c)] show the highest equivalent Abbe numbers due to the shorter light paths resulting from the non-alternating focal lengths.

The optimum selection for pure refractive hyperchromatic systems involves a first lens with negative focal lengths, followed by elements with alternating signs in the focal lengths. In contrast to our prior work [9], we succeeded in reducing the equivalent Abbe number from 2.07 to 0.987. This achievement underscores the considerable potential inherent in the use of two sequentially arranged refractive hyperchromats.

ZnSe is rarely used as a lens material due to its problematic processability, attributed to the presence of harmful selenium, and significant transmission losses in the blue visible spectral range. To address these issues, Configuration 4 was optimized as a practical alternative using SF66, which is known for its low Abbe number among commonly used inorganic glasses. This optimization process employs identical initial parameters and merit function settings. Figure 3 shows as Configuration 5 the optimized system parameters [Fig. 3(a)], the linear behavior of the axial chromatic shift [Fig. 3(b)], and the ray trace for this reconfigured setup [Fig. 3(c)].

 

Fig. 3. (a) Parameters of optimized common pure refractive systems with materials, focal lengths (f), interlens distances (D), axial chromatic shift ($\Delta {a^\prime}$), and equivalent Abbe number (${V_A}$). (b) Illustration of the wavelength dependent axial focal shift. (c) Cross section of calculated ray trace after optimization for three wavelengths.

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Although SF66 has an Abbe number of 20.89, which is more than twice that of ZnSe, the equivalent Abbe number of 2.5 achieved is particularly impressive. This indicates that the system enables axial chromatic spreading that is more than eight times stronger than that of a single SF66 lens. In addition, the wavelength-dependent focal shift exhibits a linear behavior, as shown in Fig. 3(b). This optimization emphasizes that even for widely used lens materials, the introduction of two cascaded hyperchromats has significant potential to dramatically reduce the equivalent Abbe number of refractive systems.

4. RAY OPTICAL OPTIMIZATION OF HYBRID DOUBLE HYPERCHROMATIC SYSTEMS

For the investigations of hybrid systems, identical initial values, variable parameters, and boundary conditions were employed as those used for the refractive counterparts. To model the diffractive optical element, a specific surface type (binary 2) provided by the optical design software is employed [14]. Hereby, the diffractive elements were characterized by a normalization radius of 100 mm and a variable quadratic phase term, specifically defined for the first diffraction order. Table 2 presents the four most favorable configurations for each possible position of the diffractive element, with the optimized focal lengths $f$, the quadratic phase terms ${{\rm p}^2}$, and lens distances D, their absolute equivalent Abbe numbers ${V_A}$ and the axial chromatic shift $\Delta a^\prime$.

For all four configurations, very small equivalent Abbe numbers are obtained, which are just a bit above or significantly below 1. The minimum equivalent Abbe number (${\sim}{0.65}$) is achieved in Configuration 7, featuring positive refractive lenses as the first and third elements, with a negative diffractive optical element (DOE) positioned between them and a negative refractive lens at the end. This approach results in an equivalent Abbe number that is more than five times lower than the Abbe number of a single diffractive element. An interesting and noteworthy observation is that the DOE, regardless of its position, should always have a negative optical power with a divergent character. Furthermore, as for the pure refractive systems discussed in the previous section, the hybrid systems with alternating signs of the focal lengths obtained the lowest equivalent Abbe numbers. When observing the configurations with the lowest equivalent Abbe numbers (7 and 8), it becomes evident that it is useful to place the DOE with a negative focal length between the refractive elements and not at the front or the end of the system. The ray traces that represent the color separations in the four different configurations are shown in Fig. 4.

Table 2. Parameters of Optimized Hybrid Systems^a

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Fig. 4. Cross sections and ray traces for the four hybrid cascaded hyperchromatic configurations after optimization. Configurations 6 to 9 are assigned to the designations (a) to (d).

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In general, the ray traces for the hybrid systems exhibit a behavior similar to that of purely refractive systems. Also in hybrid systems, the wavelength spreading is the largest, and the lowest equivalent Abbe numbers are achieved for configurations in which rays travel the longest paths through the system, as observed in Figs. 4(b) and 4(c). Consequently, alternating signs of focal lengths are essential for these systems. The DOE and the highly dispersive refractive elements that introduce and control the color separation have the most significant effect on the equivalent Abbe number. Because DOEs exhibit inverse dispersion behavior compared to refractive elements, highly dispersive refractive and DOEs require opposing signs in the focal lengths to enhance spectral decomposition. Low dispersive refractive lenses are employed to maximize light transmission and compensate spherical aberration. In total, Configuration 7 as the best designed system achieved a 30% reduction compared to the results of our earlier work, where a minimum equivalent Abbe number of 0.93 for a two-lens system has been obtained.

In order to demonstrate achievable equivalent Abbe numbers for hybrid systems using standard lens materials the optimization of Configuration 7—having the lowest equivalent Abbe number—was reexamined. In this iteration, similar to the previous section on purely refractive hyperchromats, ZnSe has been replaced by the more widely used material SF66. Figure 5 outlines the parameters [Fig. 5(a)], linear focal shift [Fig. 5(b)], and ray trace of the optimized system with these adjustments [Fig. 5(c)].

 

Fig. 5. (a) Parameters of optimized common hybrid system with materials, focal lengths (f), quadratic phase term (${{\rm p}^2}$), interlens distances (D), axial chromatic shift ($\Delta {a^\prime}$), and equivalent Abbe number (${V_A}$). (b) Illustration of the wavelength dependent axial focal shift. (c) Cross section of calculated ray trace after optimization for three wavelengths.

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The results shown in Fig. 5 emphasize a distinct axial chromatic spreading with a clear linear behavior. In the ray trace shown in Fig. 5(c), the system exhibits a similar structure to Configuration 7, but with a notable difference: The optimization reveals that the marginal rays are not directed close to the lens edges, thus not utilizing the available diameter. This limitation prevents achieving smaller focal lengths and lens curvatures, which would otherwise introduce spherical aberrations. As a result, while the equivalent Abbe number is higher than in Configuration 7, it is still more than half as small as the absolute Abbe number of a single diffractive lens. This demonstration highlights the potential of combining a hybrid and a refractive hyperchromatic lens to simultaneously reduce the equivalent Abbe number and achieve axial chromatic shift. This is true for both specialty lens materials such as ZnSe and standard glasses such as SF66.

5. CONCLUSION

The primary focus of this study was to explore the feasibility of using two cascaded hyperchromats to create optical systems characterized by exceptionally low equivalent Abbe numbers and distinct linear axial spectral decomposition. This investigation encompassed both purely refractive and hybrid configurations, comprising a combination of refractive and diffractive elements. Through detailed optical design, various optical quantities have been investigated, taking into account geometric constraints. The simulations showed that, for purely refractive systems, an optimal strategy involves the alternation of collective and divergent lenses to achieve an exceptionally large axial chromatic dispersion. In the case of hybrid systems, the key factors are the choice of the position for the DOE and the selection of the focal lengths. To attain extremely low equivalent Abbe numbers, it is crucial for lens two or lens three to be a DOE with a negative focal length and that all elements use alternating signs in focal lengths. Remarkably low values were achieved, with purely refractive systems reaching an absolute equivalent Abbe number of 0.983, less than half the minimum value reported in our previous work [9]. Similarly, hybrid systems achieved 0.65—more than four times lower than the absolute Abbe number of a single diffractive element. Furthermore, the demonstration of systems using only conventional materials resulted in an equivalent Abbe number of 2.5 for the pure refractive configuration and 1.4 for the hybrid one, demonstrating their relevance for commercial applications. The investigations and optimizations were conducted on an exemplary system with a specified total length of 100 mm and an entrance pupil diameter of 9 mm. It is essential to note that the obtained equivalent Abbe numbers were contingent on specific boundary conditions. By adjusting these conditions, such as increasing the total axial length of the systems or incorporating aspheric lens shapes, it becomes conceivable to achieve even lower equivalent Abbe numbers. Additionally, it is crucial to mention that the systems presented in this paper were optimized to the limits of these constraints. As a result, the systems exhibit high sensitivity, where even small deviations, such as thickness differences of a few micrometers, have a negative impact on performance. To achieve wider manufacturing and positioning tolerances for conventional optics, the system structures must be adapted, which is associated with higher equivalent Abbe numbers. In summary, the outcomes of this work are highly promising for enhancing the performance of optical applications founded on the operational principle of axial spectral decomposition, effectively transcending previous limitations in axial chromatic spreading.