1. INTRODUCTION

Since the 16th and 17th century when the first telescopes and microscopes were established, scientists and engineers have constantly tried to develop concepts to avoid chromatic aberrations. Over time, optical systems such as achromats and apochromats were developed, which allow for a drastic reduction of especially longitudinal chromatic aberrations. However, the reverse approach, specifically optical systems with an extended axial color spreading, can also be used for a number of interesting optical applications. Typically, such “hyperchromats” are used for surface profiling [1,2] and thickness measurement [3,4], for chromatic confocal microscopy [5,6], for endoscopy [7], and also for hyperspectral sensing [8], to name only a few.

The challenging derivation of optical design concepts maximizing the longitudinal chromatic aberration has already been investigated in different studies. Hillenbrand et al. [9], for example, provided general aspects for a design strategy to meet main criteria for hyperchromatic lenses. In detail, the solutions for a specific hybrid concept, combining a diffractive and a refractive lens as well as a diffractive doublet, were presented and compared. The optimal selection of dispersion properties of the lenses involved and the ideal distribution of their optical powers that maximize the longitudinal chromatic aberration were not investigated. Novak and Miks [10] describe hyperchromats with a linear dependence of longitudinal chromatic aberration on wavelength. Zhang et al. [11] introduce “initial structures” for refractive lens doublets that offer a large dispersion range and are therefore suitable for being used in a chromatic confocal sensor. Here, the Abbe numbers of the lens materials are only qualitatively considered, and the material choice remains limited to conventional, “preferred” inorganic glasses. In [12], a hybrid diffractive-refractive optical lens doublet is combined with an additional refractive lens, allowing focal tunability with a strong longitudinal (hyperchromatic) intensity spreading. A simple equation for the calculation of the effective (“equivalent”) Abbe number of the lens combination is established, which includes the refractive powers of the components involved, but its dispersion properties are not regarded.

The research presented here begins by using a simple paraxial approach in which both optical power and dispersion properties of the involved elements are systematically varied to maximize the longitudinal chromatic aberration of a lens doublet. Here, the longitudinal chromatic aberration is expressed as the inverse of the system’s “equivalent” Abbe number. Purely refractive lens combinations as well as refractive-diffractive hybrid systems are considered. Additionally, it is essential that optical materials with exceptional dispersion properties, such as nanocomposites [13] or selected compound semiconductors, are taken into account. The paraxial approach does not provide a unique minimum for the equivalent Abbe number, i.e., any large axial color splitting is mathematically possible. However, very small equivalent Abbe numbers require extreme conditions for splitting the optical power between the involved components. In order to determine the actually implementable configurations for extremely small equivalent Abbe numbers, systematic ray-trace analysis was carried out, taking into account geometric constraints for the lens curvatures. Furthermore, complete and continuous dispersion curves are used, and therefore the simplified characterization of dispersion by using solely the Abbe number of components is not necessary.

2. PARAXIAL FORMULATION OF THE EQUIVALENT ABBE NUMBER FOR REFRACTIVE AND HYBRID REFRACTIVE-DIFFRACTIVE LENS DOUBLETS

According to various authors [12,14–16], the image side longitudinal chromatic aberration $\delta a^{\prime}$ of a lens doublet for an infinite object width ($a = - \infty$) can be calculated using the following equation:

Here, $\varphi$ represents the optical power of the combined system, ${\varphi _1}$ and ${\varphi _2}$ are the optical powers of the individual components, and ${\nu _1}$ and ${\nu _2}$ are their Abbe numbers. The optical power is the reciprocal of the respective focal length, and $\delta a^{\prime}$ denotes the distance between the foci of the short and long reference wavelengths.

The optical power of the lens doublet can be calculated as the sum of the partial contributions

By normalizing the longitudinal chromatic aberration $\delta a^{\prime}$ to the focal length ${f}^\prime$ of the combined system, the “equivalent” Abbe number $\tilde \upsilon$ can be introduced as follows:

The “equivalent” Abbe number, sometimes also named the “effective” Abbe number, corresponds to the Abbe number of a hypothetical glass of a thin reference single lens that has the same longitudinal chromatic aberration as the combined optical system. An optical system is referred to be “hyperchromatic” when the equivalent Abbe number is smaller than the Abbe number of all the individual elements, and the longitudinal chromatic aberration of the entire system is therefore greater than that of each individual component. From the combination of Eqs. (1)–(3) follows:

This simple paraxial approach allows for a systematic analysis of the longitudinal chromatic aberration of a doublet with a fixed total optical power and with varying partial optical powers, depending on the Abbe numbers of the components involved.

The Abbe number ${\nu _{{{\rm ref}\_d}}}$ for the refractive lenses is determined by the refractive index of the lens material at selected wavelengths

Here, the indices characterize specific wavelengths (${\lambda _d} = {587.5618}\;{\rm nm}$, ${\lambda _F} = {486.1327}\;{\rm nm}$, and ${\lambda _C} = {656.2725}\;{\rm nm}$). The Abbe numbers of inorganic glasses used as the preferred lens materials are approximately between ${20} \lt {\nu _{{{\rm ref}\_d}}} \lt {100}$. However, it is important to note that materials showing much smaller Abbe numbers do exist. Particularly interesting materials are, for example, zinc monoxide (ZnO, ${\nu _{{{\rm ref}_d}}}\;({\rm ZnO}) = {12.4}$), titanium dioxide (${{\rm TiO}_2}$, ${\nu _{{{\rm ref}_d}}}\;({{\rm TiO}_2}) = {12.2}$), or indium tin oxide (ITO, ${\nu _{{{\rm ref}_d}}}\;({\rm ITO}) = {8.2}$) [17]. Unfortunately, only some of these materials are available as bulk materials for lenses. The other materials can usually only be used as layers. Zinc selenide (ZnSe) with a very small Abbe number of 8.1 [18] is available as a lens material and is therefore explicitly included in the following considerations.

The other materials listed cannot be used as lens materials directly. However, it is possible to embed these materials into a polymer matrix in the form of nanometer-sized particles. These nanocomposites allow for the adjustment of physical properties that lie between the values of the fillers and the matrix material. For optical applications, [19] describes the tailoring of the refractive index and the Abbe number by the choice of the matrix material and nanoparticles and the adjustment of the respective volume fraction. For our studies, we include the optical properties of $n({\rm PS}/{\rm ITO}) = {1.7}$ and ${\nu _{{{\rm ref}_d}}}({\rm PS}/{\rm ITO}) = {10.3}$, which can be derived for a specific hypothetical mixture of the polymer polystyrene (PS) and ITO-nanoparticles [13]. It should be noted that optical nanocomposites offer a significant number of potential applications but are currently not established in commercially available optical instruments. However, the prospect of these materials being available in the future justifies their inclusion in this contribution.

Figure 1 shows an Abbe diagram in which optical materials are positioned as a function of their refractive index ${n_d}$ and Abbe number ${\nu _d}$. The materials specifically considered in this contribution are represented by enlarged symbols. For orientation, a large selection of typical inorganic glasses is also included (small spots). ZnSe, ITO, and the PS-ITO nanocomposite were included as highly dispersive materials, and ${\rm CaF}_2$ was selected for its very low dispersion characteristic.

 

Fig. 1. Abbe diagram. The specific materials considered in this contribution are represented by enlarged symbols. For comparison, the small circular spots represent a selection of typical inorganic glasses.

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In addition to refractive lenses, diffractive elements are also useable in hyperchromats. Diffractive lenses are characterized by an Abbe number, which is only dependent on the reference wavelengths

Of particular interest is the very small absolute value of the diffractive Abbe number which implies a very large color splitting, and its negative sign.

In the following, the effect of the different lens combinations on the equivalent Abbe number of a lens doublet will be investigated by applying Eq. (5), first discussing purely refractive systems and then hybrid diffractive-refractive combinations. As a reference, a total focal length of ${f}^\prime = {100}\;{\rm mm}$, i.e., a total optical power of $\varphi = {0.01}\;{{\rm mm}^{- 1}}$, is assumed for all discussed cases.

A. Equivalent Abbe Number for Refractive Doublets

In the first step to investigate the dependency of the equivalent Abbe numbers of a purely refractive system on the different influencing variables, the Abbe number of the first component was fixed. Specifically, we chose the dense flint glass N-SF66 with a low Abbe number of 20.9 to achieve a high dispersion for the overall system.

Figure 2(a) shows a set of different curves for the equivalent Abbe numbers calculated according to Eq. (5). For each curve, the equivalent Abbe number is plotted as a function of the focal length of the first component of the refractive doublet. The focal length of the first component can take both positive and negative values. The individual curves differ by Abbe numbers assumed for the second component, which vary between 30.5 and 91. The specific values of the selected Abbe numbers correspond to commercially available glasses of the glass catalog. In addition to the doublets of inorganic glasses, the diagram also includes curves for special cases in which the highly dispersive materials ZnSe and ITO/PS-composite are each combined with the low dispersive ${{\rm CaF}_2}$.

 

Fig. 2. Paraxially calculated equivalent Abbe numbers for purely refractive doublets as a function of the focal length of the first component. (a) Abbe number of the first lens fixed [${\nu _1}\;({\rm N} {-} {\rm SF66}) = {20.9}$] and varying Abbe numbers for the second material. (b) Abbe number of the second lens fixed [${\nu _2}\;({\rm N} {-} {\rm SF66}) = {20.9}$] and varying Abbe numbers for the first material.

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The cases where the equivalent Abbe number tends to ${\pm}\infty$ correspond to achromatic systems. For very large positive and negative focal lengths of the first component, the second component dominates the total optical power of the system so that in these cases the equivalent Abbe number of the doublet converges towards the Abbe number of the second element. All displayed curves pass the origin of the coordinate system, showing that hyperchromatic combinations, i.e., very small equivalent Abbe numbers, correlate with small absolute values for the focal lengths of the first component. In this range, the individual curves can essentially be distinguished by their slope. The comparison of the different inorganic glass combinations shows that the curves become flatter as the Abbe number of the second component increases. The smallest gradient is achieved for the special cases when a highly dispersive material (ZnSe or ITO/PS) is combined with a very low dispersive material (${{\rm CaF}_2}$). A small slope offers the advantage that a selected small equivalent Abbe number can already be achieved for less tense focal length requirements.

Figure 2(b) shows the corresponding result for the reversed configuration. Now, for the inorganic glass combinations the Abbe number of the second component is fixed to ${\nu _2} = {20.9}$, and the Abbe number of the first element varies between 30.5 and 91. Also, for both special cases, the materials were interchanged between the first and the second component. Again, for the achromatic case, the equivalent Abbe numbers tend to ${\pm}\infty$. For large absolute values of the focal length of the first element, the equivalent Abbe numbers also now converge to the value of the second Abbe number (20.9 and 10.3 and 8.1, respectively). Again, hyperchromatic behavior is associated with a small absolute focal length, and a particularly advantageous weak slope can be achieved for doublets in which materials with the smallest possible Abbe number and the largest possible Abbe number are combined.

 

Fig. 3. Paraxially calculated equivalent Abbe numbers for hybrid doublets as a function of the focal length of the first component. (a) DOE as the first component and varying Abbe numbers for the second lens. (b) DOE as the second component and varying Abbe numbers for the first lens.

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B. Equivalent Abbe Number for Hybrid Diffractive-Refractive Systems

In the next step, the equivalent Abbe number for hybrid diffractive-refractive combinations is considered. The total focal length of the doublet is again fixed to 100 mm. Initially, it is assumed that the first component is a diffractive optical element (DOE), and the second is formed by a refractive lens. Figure 3(a) shows the calculated dependencies where again the different curves correlate with the varying Abbe numbers of the refractive lens. In the range of positive focal lengths of the DOE, significant differences between the curves for the different Abbe numbers of the refracting element are observed. Specifically, the achromatic borderline cases are reached for small Abbe numbers of the refractive component already at essentially smaller positive focal lengths of the DOE than for large Abbe numbers.

To achieve a hyperchromatic performance, very small focal lengths for the first component are again a decisive factor. Small absolute values for the equivalent Abbe number can be achieved for both positive and negative small focal lengths of the first component. In the hyperchromatic region, close to the coordinate origin, the curves are much flatter than for comparable purely refractive combinations. The lowest slope, and thus the smallest equivalent Abbe numbers, are obtained for combinations in which the refractive component has the smallest Abbe number. For completeness, Fig. 3(b) shows the reversed situation with the refractive lens on the first position and the diffractive element as the second component of the doublet. It can be seen that the curves are in close proximity to each other, only the result for very small Abbe numbers appears slightly separated. For very large positive and very large negative focal lengths of the first element, all curves asymptotically approach the limit of the value of the diffractive Abbe number. The hyperchromatic cases again correspond to small absolute values for the focal lengths of the first element. The different curves are close to each other, with maximum dispersion (minimum equivalent Abbe number) for the material with the lowest Abbe number of the refractive component.

3. COMPREHENSIVE RAY OPTICAL SIMULATIONS OF HYPERCHROMATIC DOUBLET SYSTEMS

In the preceding studies on the equivalent Abbe number of doublet systems, only paraxial equations with the optical powers of the involved components as generalized parameters were used. The very different forms that lenses of the same focal length can take have not been considered thus far. In the non-paraxial region, however, it has to be considered that for the same focal length, the characteristic data of the lens surface (e.g., the radius of curvature of a spherical lens) and shape and size of the boundary (e.g., circular with a certain diameter) significantly affect the induced aberrations. Furthermore, under real conditions, it must be taken into account that not all theoretically calculable geometric configurations can also be implemented in practice.

In the following, systematic ray-trace investigations are carried out with the aim of developing practical realizable systems and determining their achievable optical characteristics and in particular to determine the limits for the longitudinal chromatic aberration. In addition to the specific geometric quantities of the two lenses of the doublet, such as the radii of curvature and the diameters, the complete dispersion curves for the selected materials are also considered for the ray-trace analyses.

For the ray-trace investigations we chose an approach that allows comparison to the paraxial results presented before. Again, a total focal length of 100 mm is specified for the doublet system. To achieve small focal lengths for the individual components, as required for small equivalent Abbe numbers, small radii of curvature are necessary. Simultaneously, the aberrations of lenses with small radii of curvature are significantly affected by the aperture diameter. Here, a constant single lens diameter of 10 mm was chosen for the following simulations, which correlates to a numerical aperture of 0.05 at the design wavelength for the doublet. A total focal length of 100 mm in combination with a lens diameter of 10 mm results in a ${f}$-number of 10.

For each doublet variant, namely purely refractive or hybrid diffractive-refractive combinations, the focal lengths of the individual components and the optical materials with their dispersion properties are varied. In particular, for given materials of the doublet, the focal length of the first component is varied in steps. The focal length of the second component results from the requirement for a total focal length of 100 mm. The challenge here is to adequately balance the optical power between the individual surfaces so that extreme radii of curvature are avoided and practical implementation remains realistic. For illustration, Fig. 4(a) shows (schematically and simplified) the geometrical changes of the lens curvatures for the purely refractive case when varying the focal lengths of the individual components and maintaining the total refractive power. Specifically, extreme radii of curvature can be expected for small focal lengths of the first component corresponding to the hyperchromatic state. Here, it is particularly important to track feasibility and control any aberrations that occur, e.g., by analyzing wavelength dependent spot diagrams in different observation planes. The commercial optical design software OpticStudio [20] was used for the ray-tracing analyses.

 

Fig. 4. (a) Schematic illustration of different single-lens geometries of a doublet with constant overall optical power. Numbers (above) refer to the focal length of the first component. (b) Schematic representation of the approach for determining the equivalent Abbe number for each particular system. In order to optimize the spot diagrams for the different wavelengths (${\lambda _d},\;{\lambda _F}$, and ${\lambda _C}$) in their respective target planes, the radii of curvature of the components involved are systematically varied.

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To analyze the equivalent Abbe number for the different focal length combinations, we chose an approach described below and shown schematically in Fig. 4(b). First, the same wavelengths as before (${\lambda _d},\;{\lambda _{F}}$, and ${\lambda _C}$) are set as reference. For these wavelengths, the imaging properties of the varying doublet systems are considered. Therefore, two spherical lenses are used as a starting point in each case, assuming that the radii on the inner surfaces facing each other are equal. In the hybrid case, the diffractive structure is assumed to be on a flat surface, which potentially simplifies a lithographic fabrication. In the image space behind the doublet, different target planes result for the three reference wavelengths in which a minimum diameter for the spot diagram is achieved. To optimize the doublet systems, a merit function is defined, which aims at minimizing the spot radii in the target planes for the respective wavelengths. The optimization process is based on the systematic variation of the radii of curvature of the components involved, with both the focal lengths of the individual components and the overall focal length of the doublet maintaining their specified value. Finally, the distance between the reference planes for minimum and maximum wavelength is used as the measure for longitudinal chromatic aberration $\delta a^{\prime}$. According to Eq. (3), the reciprocal of $\delta a^{\prime}$ related to the focal length describes the equivalent Abbe number. In addition, the resulting minimum spot radii are determined for each of the three reference wavelengths, and the Airy radii are recorded accordingly.

A. Ray-Trace-Based Determination of the Equivalent Abbe Number for Refractive Systems

As before in the paraxial approach, in the first step, a material with a small Abbe number is specified for the first component of the doublet, and the material of the second component is varied. Again, the highly dispersive N-SF66 is combined with various other inorganic glasses, and, to achieve extreme values for the equivalent Abbe number, doublets comprising a ZnSe or a PS/ITO-composite lens and a ${{\rm CaF}_2}$ lens are also considered. As a result, Fig. 5(a) shows the equivalent Abbe numbers that were determined by the method described before as a function of the focal length of the first component for the different material compositions. The diagram is limited to the hyperchromatic region near the coordinate origin since this is the main focus of this investigation.

 

Fig. 5. Equivalent Abbe numbers for purely refractive doublets as a function of the focal length of the first component determined by a ray-trace-based approach. (a) Abbe number of the first lens fixed and varying Abbe numbers for the second material (first lens either N-SF66, ZnSe, or PS/ITO-composite). (b) Abbe number of the second lens fixed and varying Abbe numbers for the first material. All solid curves with symbols represent the ray-trace results, and the dashed lines display the boundary cases for the paraxial calculations.

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The circles in the diagram represent the determined results of the equivalent Abbe number for the individual focal lengths of the first component. The connecting lines are for better visualization. In addition, the diagram also includes the boundary cases from the previously calculated set of paraxial curves (dotted lines without symbols). The comparison between ray-trace-based curves and paraxial results show a good agreement. In particular, all curves have a similar form, and the calculated ray-trace-based curves lie between the paraxial boundary cases.

For all curves it can be observed that with decreasing absolute values for the focal length of the first lens, the equivalent Abbe number tends to zero. Particularly small equivalent Abbe numbers are achieved for systems with the smallest possible Abbe number for the first component and the largest possible Abbe number for the second component. Explicitly, for the combination of ZnSe and ${{\rm CaF}_2}$, we achieve an equivalent Abbe number of 2 for a focal length of the first component of $+20\;{\rm mm}$. With a corresponding focal length of ${-}{20}\;{\rm mm}$, even an equivalent Abbe number of ${-}{1.2}$ is achieved, an absolute value far below the one of a single diffractive lens.

Figure 5(b) displays the results for reverse cases with either N-SF66 as the material for the second component and different inorganic glasses for the first, or ZnSe and PS/ITO, respectively, for the second lens and ${{\rm CaF}_2}$ for the first. The corresponding paraxial boundary cases are also plotted and again a good agreement with the paraxial calculations in Section 2.A can be observed. For the combination of ZnSe and ${{\rm CaF}_2}$, the smallest equivalent Abbe numbers are again obtained, and the curve shows the smallest slope. Here, for a focal length of ${-}{20}\;{\rm mm}$ of the first component, an equivalent Abbe number of 1.4 is achieved, and for a corresponding positive focal length of ${+}{20}\;{\rm mm}$, we obtained an equivalent Abbe number of ${-}{1.2}$.

B. Ray-Trace-Based Determination of the Equivalent Abbe Number for Hybrid Diffractive-Refractive Systems

Figure 6 shows the equivalent Abbe numbers resulting from ray-trace simulations for both hybrid combinations. Figure 6(a) shows the DOE as the first element followed by the spherical refractive lens, while Fig. 6(b) shows the reversed configuration, with the refractive element first, followed by the DOE. Again, the materials of the refractive component are varied. For large absolute values of the focal length of the first element, high agreement with the paraxial results is found (not shown). However, for very small absolute values of the equivalent Abbe numbers, significant differences to the paraxial view can be observed. The limiting paraxial cases for the combination of ZnSe and ${{\rm CaF}_2}$, respectively, with the DOE are depicted as dotted lines without symbols.

 

Fig. 6. Equivalent Abbe numbers for hybrid doublets as a function of the focal length of the first component, determined by ray-tracing. (a) DOE as the first component and varying Abbe numbers for the second material. (b) DOE as the second component and varying Abbe numbers for the first lens. All solid curves with symbols represent the ray-trace results, and the dashed lines display the boundary cases for the paraxial calculations.

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In case of Fig. 6(a) (the DOE first), the equivalent Abbe number for the small negative focal lengths of the first component does not approach zero, but it reaches a horizontal limit, which is dependent on the Abbe number of the refractive component. For large Abbe numbers larger positive values are also obtained (e.g., for ${\rm CaF}_2$, $\tilde \upsilon \approx 3$), and small Abbe numbers correspond to small values (e.g., for ZnSe, $\tilde \upsilon \approx 1.8$). The limiting effect is due to the software optimization procedure, which no longer succeeds in keeping the total focal length of the system at 100 mm when the absolute value of the focal length of the first component is reduced to small values. However, very small absolute values for the equivalent Abbe number are not obtained for negative focal lengths of the first components.

In contrast, regarding hyperchromatic behavior, the range of small positive focal lengths is much more interesting. In this region, the curves for the different materials are close together. With a reduction of the focal length of the first component, the equivalent Abbe number also decreases, whereby extremely small absolute values are achieved.

For completeness, the results for the hybrid combination with first a refractive component followed by a DOE are shown in Fig. 6(b). Extremely small negative equivalent Abbe numbers are obtained for small absolute values of the negative focal lengths of the first component. Whether these extremely small equivalent Abbe numbers can actually be achieved with sufficient optical quality will be investigated in the following section.

 

Fig. 7. (a) Cross section of a calculated ray-trace for one exemplary configuration after optimization for three wavelengths. (b) Spot diagrams in the target planes ${ C}, { d}$, and ${F}$ (columns) for the three reference wavelengths (rows).

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C. Ray-Trace-Based Evaluation of the Optical Quality of Hyperchromatic Doublets

As mentioned before, the optical quality is examined by evaluating the respective spot diagrams of each setup in OpticStudio. In the preceding, it has been shown that for extremely small absolute values of the equivalent Abbe numbers of the doublet, very small absolute values are required for the focal lengths of the individual components. With decreasing focal length of the components, the influence of aberrations increases, resulting in a reduction of optical image quality. In the following, the limits of the achievable optical qualities are analyzed by examining the spot diagrams in different target planes. Hereby, a target plane, indicated with ${C},{ d}$, and ${F}$ for the reference wavelengths, is the plane in which the spot for the respective wavelength shows a minimum diameter.

Figure 7(a) displays a cross section of the calculated ray-trace for one exemplary configuration after optimization for the three wavelengths. The respective spot diagrams are shown in Fig. 7(b) with the columns attributed to the target planes ${C},{ d}$, and ${F}$. For each target plane, the spot diagrams are calculated for all three wavelengths. The rows of Fig. 7(b) correspond to the three wavelengths. For each wavelength, the diagram showing the smallest spot distribution corresponds to the respective target plane. For completeness, the non-diagonal elements show the spot diagrams for wavelengths that do not correspond to the respective target planes. In these cases, the magnitude of the spots is extremely large (see scale). For the diagonal elements, the diameter of the respective Airy-disk is additionally indicated as a measure for the diffraction limit (black circle).

To evaluate the imaging quality of the respective configuration and focal length of the individual component, the RMS diameter of the spot diagram is compared to the Airy-disk diameter. The analysis was performed for both purely refractive and hybrid systems, considering all material combinations discussed above. In the following, only results for configurations and material combinations are presented for which extreme equivalent Abbe numbers are achieved.

Table 1 shows the result for four selected configurations. Both purely refractive configurations combine ZnSe and ${{\rm CaF}_2}$, with the first and the second element interchanged. The hybrid systems combine a DOE with a refractive ZnSe element. Colored symbols are used to illustrate the imaging quality. Green symbols represent configurations for which the RMS spot diameter is smaller than the diameter of the Airy-disk. If the RMS spot diameter measures between one and three times the Airy-disk diameter, yellow symbols are used. Configurations for which an RMS spot diameter larger than three times the Airy-disk diameter is determined, are marked in red.

Table 1. Spot Quality of Different Doublet Configurations (Purely Refractive or Hybrid) for Small Equivalent Abbe Numbers^a

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Table 2. Radii of Curvature for the Configurations Offering Both Very Small Equivalent Abbe Numbers and Suitable Imaging Quality^a

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For each of the four configurations, the imaging quality determined is shown in three columns representing the result in the target planes of the reference wavelengths. Additionally, the focal lengths of the first component (left) and the corresponding equivalent Abbe numbers (right) are listed. The focal lengths for the first component correspond widely for the four configurations. Deviations occur only in the range of very small absolute values for the focal lengths. These deviations are due to the optimization process used by the optics design software. For each configuration, the system’s overall focal length of 100 mm is kept constant while the image quality is simultaneously optimized in the three target planes. Due to these constraints, it is not always possible to fix the focal length of the first component to the specified value.

In the table, the doublet systems that offer high imaging quality (green symbols) and simultaneously a very small equivalent Abbe number are of particular interest. According to this criterion, the best values for the refractive system ${\rm ZnSe} {-} {{\rm CaF}_2}$ are $\tilde \upsilon ({{f_1} = - 75\;{\rm mm}}) = 8.0$ and $\tilde \upsilon ({{f_1} = 35\;{\rm mm}}) = - 3.6$. For the inverted configuration ${{\rm CaF}_2} {-} {\rm ZnSe}$, values of $\tilde \upsilon ({{f_1} = - 35\;{\rm mm}}) = - 2.5$ and $\tilde \upsilon ({{f_1} = 35\;{\rm mm}}) = 4.8$ are obtained accordingly. This means that the smallest absolute equivalent Abbe number of a refractive system, which still allows a suitable imaging quality, is smaller than the Abbe number of a single diffractive lens and does not suffer from strongly varying, wavelength-dependent diffraction efficiency. For the hybrid system DOE-ZnSe, the best values for the equivalent Abbe number of $\tilde \upsilon ({{f_1} = - 250\;{\rm mm}}) = - 3.8$ and $\tilde \upsilon ({{f_1} = 50\;{\rm mm}}) = 1.0$ are obtained. Among all considered configurations with sufficient imaging quality, the smallest equivalent Abbe numbers result for the inverted hybrid combination ZnSe-DOE with $\tilde \upsilon ({{f_1} = - 50\;{\rm mm}}) = 0.4$ and $\tilde \upsilon ({{f_1} = 41.4\;{\rm mm}}) = - 2.0$. The value 0.4 is a factor of ${\gt} 8$ smaller than the Abbe number of a single diffractive lens, and therefore also causes an eightfold spectral spreading.

For the final discussed configurations, which offer both very small equivalent Abbe numbers and suitable imaging quality, Table 2 contains the data on the corresponding radii of curvature of the refractive elements involved and on the phase distributions of the DOEs. In detail, for the quantification of the DOE phase, the quadratic phase term (${p^2}$-value) of the OpticsStudio surface type “binary 2” is used with a normalization radius of 10 mm.

4. CONCLUSION

In conclusion, hyperchromatic doublet systems have been systematically studied in order to obtain extreme values for the equivalent Abbe numbers. Both purely refractive doublets and hybrid systems combining diffractive and refractive components have been considered. First, basic paraxial studies were performed to pre-select materials with optimal dispersion properties and to find target values for the optical powers of the constituents. The simulations showed that in purely refractive systems, one material with a very low Abbe number and a second material with a very high Abbe number must be combined to achieve an extremely large axial chromatic dispersion. For hybrid systems it was found that the refractive component should have the smallest possible Abbe number. Following the preliminary investigations, systematic ray-trace analyses were carried out in order to select from the configurations with, in principle, extremely small equivalent Abbe numbers, those that are also practically implementable. In particular, geometric constraints on lens curvatures were considered and complete continuous dispersion curves were included. As extreme values for systems with appropriate imaging quality, an equivalent Abbe number of $\tilde \upsilon = \; - 2.5$ is obtained for the purely refractive variant and a value of $\tilde \upsilon = \;0.4$ for the hybrid case. The presented investigations and optimizations were performed on an exemplary system with a specified focal length of 100 mm. In general, the basic strategic approaches for the development of systems with large axial dispersion can also be applied to other system specifications. It should also be noted that the limits reached here result from the restriction to spherical surfaces of the lenses involved. It can be assumed that with the use of aspheric shapes a further decrease in the equivalent Abbe number might be achieved.