1. Introduction

Optical design and the development of optical materials have been entangled ever since the first sophisticated optical instruments were brought to life [1]. This holds especially true in the field of optical glass development, where meeting optical design specifications, such as refractive indices and chromatic dispersion values, requires precise control over the glass composition. In return, material developers are presented with fundamental physical and chemical challenges, which force optical designers to develop compensation concepts.

The compensation concepts, mentioned above, involve the combination of more than one optical glass composition each of which exhibits different refractive indices and chromatic dispersion values. To bring universal practicality to this compensatory entanglement between material and design required the development of a two-dimensional representation that categorizes optical glasses according to both quantities. This 2D representation was introduced by SCHOTT in 1923 and named Abbe diagram after the physicist, entrepreneur, and founder of the CARL ZEISS Foundation, Ernst Abbe.

In the classical version of the Abbe diagram graph ([2], Fig. 1), the refractive index is represented by n_d, which is the refractive index at 587.56nm (yellow). Concerning chromatic dispersion, only primary colour is considered - in other words, the dispersion of blue and red wavelengths - and is represented by the Abbe number

 

Fig. 1. Abbe diagram according to classical definition. Optical glasses from SCHOTT AG. N-glasses: Pb- and As-free. P-glasses: low-Tg-glasses especially developed for precision moulding (Tg: glass transition temperature).

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To serve the requirements of RGB (red, green, blue) colour management, an alternative definition notated ν_e has been introduced where n_e, the index at 546.07nm (green), replaces n_d.

The filled part of the Abbe diagram represents what is feasible with optical glasses; the empty part, or the white space, what is not. The so-called white space has driven the invention of multi-element lenses, which mimic a single lens element from a virtual material in the upper left corner of the diagram.

Table 1 explains the labels (acronyms of the chemical composition in German):

Table 1. Significance of the glass type labels used in the Abbe diagram

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For a complete assessment of the optical performance of components made from glass or any optical material, however, the information provided by the Abbe diagram is not sufficient. As the refractive index does not linearly depend on wavelength, an additional quantity describing its nonlinearity is required for a complete picture of chromatic dispersion.

For this purpose, the relative partial dispersion P has been introduced. Depending on the wave­length range being considered, one may choose between various definitions. The most common is:

 

Fig. 2. Relative partial dispersion.

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Another definition of relative partial dispersion (which one may also refer to when considering imaging challenges in the visible) is:

In the following, it will, first, be explained what are the classical needs of optical design and, second, how the challenges presented by these classical needs have been met by material developers within the last centuries. Particular emphasis will be placed on the work of Otto Schott and the entanglement of the optical designer and optical material developer as past and present solutions are discussed.

2. Classical needs of optical design

2.1 Suppression of spherical aberration

In general, lens elements can easily be ground and polished by a set-up of rotating devices [3], a process that has already existed in history [4]. The disadvantage of such a machine is that it can only produce spherical spheres. These, however, cause spherical aberration. Consider a plano-convex lens, see Fig. 3.

 

Fig. 3. Spherical aberration.

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Spherical aberration primarily affects rays far off the optical axis. Calculating the focal length f, i.e., the position where the ray considered crosses the optical axis, as a function of the distance h between the ray and the optical axis, one obtains up to second order [5]:

For small values of h, the second term in Eq. (4) can be neglected. However, for larger values of h, the focal length depends on h via h²⁄(2R). This is the longitudinal effect of spherical aberration. To minimize h²⁄(2R), one has to make R big. To maintain the focal length of the rays close to the optical axis, one has to compensate for this increase of R by a corresponding increase of n. Taking this into consideration, the major challenge to develop glasses with high refractive indices has been the need to suppress spherical aberration.

2.2 Correction of primary and secondary colour

With respect to spherical aberration and chromatic dispersion, lens elements from an imaginary optical material situated in the upper left corner of the Abbe diagram would be most preferable. With real optical materials, however, chromatic dispersion leads to a longitudinal colour issue as described by Fig. 4.

 

Fig. 4. Longitudinal chromatic dispersion.

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As optical materials in the upper left corner of the Abbe diagram cannot exist due to fundamental physics (see the Kramers-Kronig relations below), optical design has had to step in. Combining a positive (converging) and a negative (diverging) lens element, one may arrive at a doublet where the “red focus” and the “blue focus” coincide, i.e., where the primary axial colour issue is eliminated. A system of this optical design is called an achromate, see Fig. 5.

 

Fig. 5. Primary axial colour correction, residual secondary axial colour.

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To find a suited combination of refractive indices and Abbe numbers, some calculations are necessary [5]. The starting points are the lensmakeŕs equation Eq. (5) and the additivity rule Eq. (6) for the inverse focal lengths of thin lens elements (note that all radii which are convex to the left are counted positive; all radii which are convex to the right are counted negative):

The blue and the red focal length will be the same, if the following equation holds:

As both 1/R_1,1 and -1/R_1,2 are positive, (n_1,F – n_1,C) should be small in order not to create a large positive first term and making compensation by the negative second term less easy.

R_2,1 and -R_2,2 are both negative so that compensation of the first term is possible. At this point, it has to be decided which factor of the second term, either (n_2,F–n_2,C) or (R_2,1-R_2,2) should bear the load of compensation. A big value of (R_2,1-R_2,2) would mean a negative lens element with small radii of curvature, which would cause spherical aberration and be neither easy to manufacture nor to handle. The resulting negative focal length might be even so small that the doublet as a whole is not converging. So preferable, (n_2,F–n_2,C) is made large.

Expanding the first term of Eq. (7) by n_1,d and the second one by n_2,d leads to:

Here, f_d is the focal length according to Eq. (5) for n = n_d. Applying (8) one finds that the combination of a positive element from N-LAK8 (ν_d = 53.83) with f_d = 100mm and a negative element from N-SF6 (ν_d = 25.36) with f_d = - 212mm will be achromatic [5].

With Eq. (8) fulfilled, the primary colour issue is solved. There still may be, however, secondary colour. If one considers the partial dispersion values of N-LAK8 (P_g,F = 0.545, P_d,C = 0.3042) and the ones of N-SF6 (P_g,F = 0.6158, P_d,C = 0.2867), one finds (1) that the blue-to-yellow-dispersion is much bigger than the yellow-to-red-dispersion for both glasses. This is due to the superlinear wavelength dependence of the refractive index. One finds, second, that this superlinearity is stronger for N-SF6 than for N-LAK8. This is consistent with the fact that both glasses lie on the normal line of partial dispersion.

Therefore, even if the red-to-blue dispersion in the positive element is compensated by the red-to-blue recombination in the negative element, the red-to-yellow dispersion in the positive element is not completely compensated by the red-to-yellow recombination in the negative element. As such, the yellow focus remains closer to the lens element than the blue and red.

It is different if a positive element from N-SSK8 (n_d = 1.617783, ν_d = 49.83, P_g,F= 0.5602, P_d,C = 0.2999) is combined with a negative element from N-KZFS11 (n_d = 1.6377, ν_d = 42.41, P_g,F= 0.5605, P_d,C = 0.3000) [5]. As the P_d,C–values are equal, solving the primary colour issue will solve the secondary colour issue also. With this optical design, one obtains an apochromate; see Fig. 6.

 

Fig. 6. Primary and secondary axial colour correction.

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Note that N-KZFS11 belongs to the glasses with anomalous partial dispersion; see Fig. 2.

2.3 Transmission in the visible

To ensure optimum transmission in the visible, optical glasses have to be made from oxides, the bandgap of which is so high that only photons with an energy corresponding to a frequency in the ultraviolet (UV) will be absorbed. Such oxides are, for example, SiO₂, B₂O₃, and CaO [5]. Low band gap oxides, which act as colouring agents, such as FeO or Fe₂O₃, are carefully avoided.

From the bandgap values of the oxides involved in the composition, one may get a first estimate of the UV absorption edge of the corresponding glass. For quartz glass, the UV absorption edge will be at 160 nm [6] corresponding to a bandgap of 8.9 eV of alpha-quartz [7]. This is only true, however, in the absence of impurities. Small traces of TiO₂, for instance, will give rise to an absorption peak in the UV [8] whereas increasing concentrations of TiO₂ in the composition will shift the UV absorption edge to longer wavelengths.

For a deeper understanding of the UV absorption edge, one has to take into account the particularities of the glassy state which cause a “smearing out” of the absorption edge (Urbach-Tail [9]).

At lower frequencies or longer wavelengths, the visible (VIS) and the near infrared (NIR), and in the complete absence of impurities, the internal transmittance of an oxide glass is 100%; see Fig. 7. This transmission window closes in the near infrared because of the 2.8 µm absorption peak of hydroxyl groups in glass [10].

 

Fig. 7. Internal transmission in UV, VIS, NIR for three glasses from SCHOTT AG.

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2.4 Limitations following from Kramers-Kronig relations

The above mentioned limitations to the values of refractive index, chromatic dispersion, and transmission can be identified via the Kramers-Kronig relations [11]. Note that these relationships between macroscopic quantities are valid, independent from the further relationship between those macroscopic quantities and their microscopic counterparts. For solids like glass, the latter relationship is the Lorenz-Lorentz equation [11].

The Kramers-Kronig relations state that within linear response theory, there is an interrelation between the real part and the imaginary part for any susceptibility χ, in particular for the electric susceptibility, independent from the material (glass or polymer or …):

ɛ is the electric permittivity, ω is the angular frequency, P stands for Cauchýs principal value of the integral. With the following relation between the complex permittivity and the complex refractive index n* [5] (κ absorption coefficient, λ wavelength):

The message conveyed by Eq. (12) is that in any material there is no high refractive index without an absorbance issue and a dispersion issue. Consider, first, the numerator of Eq. (12); there has to be a non-zero κ in the vicinity of ω on the frequency scale. The tail of this absorption peak or absorption edge may affect the transmission at ω. Second, consider the denominator of (12). The closer ω is to the next ω’ with non-zero κ(ω’), the bigger both the value and the wavelength dependence of n at said ω will be.

The aforementioned intricate balance of optical design rules is what makes optical material development a real challenge. It also explains why the upper left corner of the Abbe diagram is empty and will be forever.

For practical purposes, one discretizes the integral, replaces ω with 2πc/λ and thus arrives at the Sellmeier series, a semi-empirical formula for the frequency or wavelength dependence of the refractive index, with coefficients B_i, C_i that have to be provided for each glass (and will be by any comprehensive optical glass catalog):

The continuous absorption of Eq. (12) has been replaced with a discretized absorption located at wavelength positions, the squares of which correspond to the C_i. The B_i, denote the strengths of these absorbers and incorporate the κ from above. Note that Sellmeier published Eq. (13) in 1871 [12], i.e., half a century before the Kramers-Kronig-relations were introduced.

The refractive index curves of typical optical glasses show the general tendency to be predictable with respect to the Kramers-Kronig relations: the higher the index (Fig. 8), the closer the UV absorption is to the visible range (Fig. 7), and the larger the nonlinearity of the refractive index becomes in the short wavelength part of the visible range (Fig. 8).

 

Fig. 8. Refractive index in UV, VIS, NIR for three glasses from SCHOTT AG

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The above mentioned Lorenz-Lorentz equation linking macroscopy and microscopy will be referred to below and is given here therefore [11]:

3. Corresponding glass development

Considering the Kramers-Kronig relations, one will find directives for the glass development required to meet the above classical needs of optical design. In fact, although not all of them could know what Kramers and Kronig would publish in the 1920s, the glass developers of modern times, in particular Otto Schott, followed these directives.

3.1 Transmission

At the beginning of modern times, around 1450, the Venetian (Murano) glass maker Angelo Barovier managed to manufacture clear white glass (Cristallo). He was not able to eliminate colouring impurities like iron oxide from the glass; instead, he titrated the melt with manganese oxide until decolouration was reached [13]. Manganese oxide has an absorption spectrum which is complementary to the other colouring oxides so that at the end, a constant (“grey”) absorption is reached which appears (almost) white. These glasses were of the soda/potash-lime-silicate type [13], and commonly referred to as “crown glasses” in optics originating from the special manufacturing method [14] used at that time. The so-called “crown glasses” were used in the telescopes and microscopes invented ca. 150 years later [1].

Despite progress having been made in glass melting in the meantime, the transmission issue was still quite challenging when Otto Schott and his coworkers faced the task of improving optical glass in the late 19^th century. Impurities were not yet fully under control and a comprehensive understanding of the impact of each individual oxide on transmission did not yet exist. Fortunately, the manufacture of high purity raw materials was making great progress at that time [15]. In his book on the chemical technology of glass, Otto Schott´s coworker Eberhard Zschimmer points out that the “Purissimum” materials from Merck (today: MERCK KGaA, Darmstadt, Germany) and the chemically pure silica from Heraeus (today: HERAEUS Holding GmbH, Hanau, Germany) allowed him to carry out a number of pioneering experiments and thus contribute to the intense worldwide research concerning the transmission of glass at that time [16].

First, Zschimmer checked the impact of common glass components (SiO₂, B₂O₃, Al₂O₃, Na₂O, K₂O, CaO, BaO, ZnO, PbO) on the position of the UV absorption edge finding that the network modifiers (metal oxides) would shift the absorption edge to longer wavelengths relative to those of glasses made from the pure network formers (SiO₂, B₂O₃). As one would expect today considering the bandgaps, Zschimmer found the biggest such impact for PbO.

For these experiments, Zschimmer used a spectrometer from CARL ZEISS with a fluorescent material as UV detector. Measuring or calculating the refractive index for the glasses of Table 2 one will find these glasses to be an early example of the correlation between high indices and long wavelengths for the UV edge, corresponding to Kramers-Kronig.

Table 2. UV edge of test melts (composition in wt%)

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Second, Zschimmer checked the impact of numerous colouring agents. He found even tiny amounts of iron-, manganese-, chromium-, and titanium-compounds to be particularly strong at absorbing at short wavelengths.

This research enabled the adjustment of the UV absorption edge at a suitable position for optical imaging and even the development of special UV-transparent glasses.

Today, the controlled purity of raw materials and meticulously controlled production processes allow the adjustment of impurity concentrations to the desired value [1].

3.2 Primary colour

The glasses used in early optics, for example soda/potash-lime-silicate glasses (e.g., B270 from SCHOTT AG) are positioned in the centre bottom of the Abbe diagram (B270: refractive index n_d = 1.52, Abbe number ν_d = 58.5). For primary colour correction, a second glass from the upper right of the Abbe diagram was needed.

Simplification of Eq. (13) informs the material developer of the starting point for such a glass. It is assumed that it is only the absorption at the UV edge which accounts for the refractive index in the visible range:

The derivative of n(λ) with respect to the wavelength is:

Moving λ₀ from a typical value for a crown glass, e.g. 305nm for B270 (305nm corresponds to 50% internal transmission of a sample of thickness 1.15mm), towards the boundary between the ultraviolet and the visible range, i.e. 400nm, will increase both the values of n(λ) and the wavelength derivatives of them so that the desired glass will result.

A suitable way is to take lead silicate glass as the base system. PbO·2SiO₂, for instance, has a bandgap of 3.1 eV exactly corresponding to 400nm [17]. In fact, this approach is essentially that George Ravenscroft took to develop the first clear lead silicate glass, which received a patent in 1674 [18]. Interestingly, this lead silicate was first used for tableware for which the sparkling effect due to high dispersion was a desired feature. Further, as flint stone was used as raw materials for these glasses, they have been named flint glasses [1].

The method was not used to solve the secondary colour issue until the middle of the 18^th century when the first achromates were built from crown glass and lead silicate [1]. Two names are attached to it: Chester Moore Hall, who is said to be the actual inventor, and John Dolland, who patented and started marketing telescopes with achromates around 1760 [1]. In the early 18^th century, work on achromates was continued by Joseph von Fraunhofer. “He was the first to melt from well-defined recipes and to measure refractive index and dispersion of glasses at well-defined wavelengths, the Fraunhofer lines of the solar spectrum.” [1].

3.3 Secondary colour

Despite efforts to broaden the chemical basis of glassmaking by a number of scientists including Reverend Vernon Harcourt [1], there were only five substances, for which the impact on the optical properties of glass was known in detail in the late 19^th century [19], namely SiO₂, PbO, Na₂O, K₂O, CaO. With them, however, the issue of secondary colour could not be solved. For glasses made from these components, the partial dispersion P_g,F would follow the normal line in Fig. 2.

Considering Eq. (16), one may explain the existence of the normal line like this: By moving λ₀ in the second term in the denominator of Eq. (16) to longer wavelengths, both the first and the second derivative of the refractive index with respect to the wavelength are increased. Succinctly, both main and partial dispersion increase simultaneously. As long as this is the only way to increase main dispersion, no glass pairs with different main dispersion, but equal partial dispersion are possible.

Therefore, secondary colour remained a challenge when Otto Schott appeared on the scene. From the point of view of Eq. (16), his approach was: for the flint glass negative element, he reduced the “dangerous” strong wavelength dependence of the second term in the denominator of Eq. (16) by moving λ₀ to smaller wavelengths. This alone also decreased main dispersion. Therefore, he tried to increase “B” for compensation.

Otto Schott found a suitable system among the lead borates. Lead borate glass has a larger band gap [17] and a higher packing density than lead silicate glass of the same stoichiometry. The larger band gap means that λ₀ has a smaller value. The higher packing density increases the volume fraction filled with material and thus the absorption coefficient at λ₀, which in return increases “B”. This corresponds to the Lorenz-Lorentz equation, Eq. (15), according to which the refractive index depends on the packing density and the polarizability of the atoms or ions, which in return depends on the size of the atoms or ions, see [20].

The different packing densities of lead silicate and lead borate can be obtained from the following consideration. For glassy PbO·2SiO₂, the density is 5.133g/cm³ (interpolated from values in [17]) which corresponds to a molar volume of 66.89 cm³. With the coordination being 4-fold both for silicon as well as lead in such a glass [21], and with a consistent value for the average coordination number for oxygen, one will obtain the accurate ionic radii from Shannońs coordination-number dependent “Effective Radii” [22]. With the latter, one will arrive at a packing density of 0.507.

For glassy PbO·2B₂O₃, the density is 4.368g/cm³ (interpolated from values in [17]), resulting in a molar volume of 82.98 cm³. With 50% of the boron atoms having a 3-fold coordination, the other 50% of the boron atoms having a 4-fold coordination, lead having a 6-fold coordination [23], and with a consistent value for the average coordination number for oxygen, one will again obtain the accurate ionic radii from Shannońs “Effective Radii”. With the latter, one will arrive at a packing density of 0.588.

Otto Schott put the significance of boron oxide for apochromatism in the following words [19]: “So ist in der That die Borsäure die Grundlage für alle diejenigen Flintglassorten geworden, welche eine Verminderung des sekundären Spektrums geben sollen.“ Translated into English: “So indeed, boric acid has become the basis for all those flint glass types which are supposed to provide a reduction of the secondary spectrum.”

This statement followed Otto Schott´s sys­tematic study of the optical properties of various glass compositions. The details were kept secret for a long time [16]. They are reported in Eberhard Zschimmeŕs book [16], which was first printed, but not publicly sold, in 1913, and as crazy as it might seem, translated and published in English as late as in 2013.

A typical example of the borates considered here is the borate flint S7 with 55.5mol% B₂O₃, 30.3mol% PbO, and 14.2mol% Al₂O₃ [24]. It comes close to the above composition PbO·2B₂O₃. However, to increase the chemical durability, some Al₂O₃ has been introduced in the composition. (The “S” in “S7” means that special melting is required for this glass [19].)

In parallel, Otto Schott also worked on the partial dispersion of crown glass in order to make it fit to the partial dispersion of the borate flints. He found the traditional potassium-containing crown glass composition to be superior to a corresponding sodium-containing composition. In one of his glasses, O13, the potassium content reaches 33wt% which makes this glass a perfect match for the above borate flint S7. As in the composition of S7 above, O13 also contains a few weight percent Al₂O₃ to increase chemical durability [16]. (The “O” in “O13” means that ordinary melting is sufficient for this type of glass [19].)

The optical properties of O13 and S7 are given in Table 3 [19]:

Table 3. Optical properties of some Otto Schott´s first glasses suited for apochromatic lens design

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Note that Otto Schott made reference to n_D = n(589.3nm) rather than to n_d = n(587.6nm). Consequently, n_D replaces n_d in the Abbe number given here, which is called ν_D. The partial dispersion referred to is P_F,D = (n_F – n_D)/(n_F – n_C).

Combining O13 and S7 allows an almost complete elimination of secondary colour issues.

Despite a possible Al₂O₃ admixture, chemical durability remained an issue. The chemical durability of borate glasses had been a long-term issue before. Both B₂O₃ and P₂O₅ had been known as glass formers; however, with respect to the poor chemical durability of both systems, developers hesitated to use them for their glasses [19]. Otto Schott finally solved this issue by combining boron oxide with silicon oxide in a proportion that would give chemically resistant glasses. Today, special short flints like N-KZFS11 are of the borosilicate type. The chemical resistance of this new type of glass, together with the possibility to produce low thermal expansion versions of it, soon created an ongoing high demand for technical applications. In principle, todaýs Borofloat goes back to Otto Schott.

The high potassium content of O13 makes chemical durability an issue for this glass, too. One way to solve this was the development of borosilicate crown glasses, with one of the most well-known being SCHOTT N-BK7.

3.4 Spherical aberration

Following Harcourt [25] and the German chemist Johann Wolfgang Döbereiner [26] who had already tried to replace lead with barium in the early 19^th century, Otto Schott also introduced barium in his glasses. This element has turned out to be one of the key enablers for high index, low dispersion glasses.

With respect to Eqs. (15, 16), barium is definitely a good choice. Compared to lead oxide, the band gap is quite high; it amounts to 6eV [27]. This shifts λ₀ to much lower values than those for lead silicates or borates. The inclusion of barium in glass composition should result in low-dispersion glasses; compare the above UV absorption edge values from Zschimmer.

Consider, for instance, glassy sanbornite BaO·2SiO₂, the density of which is 3.723g/cm³ [28]. Going back to Shannońs “Effective Radii” and comparing the values given there for different coordination numbers with Paulinǵs packing rules [29], one finds a consistent combination of radius and coordination number only for a 12-fold coordination of barium. With the coordination of the silicon ions being 4-fold, and a consistent value for the average coordination of the oxygen ions, one will obtain the accurate ionic radii from Shannońs “Effective Radii”. With the latter, one will arrive at a packing density of 0.589, a comparatively high number.

For both the barium crowns and the dense crowns, barium is a major component, e.g. for N-SSK8, which may be regarded as a modification of BaO·2SiO₂.

A later milestone on the way to high index, low dispersion glasses has been the introduction of lanthanum to the melt by George W. Morey [1].

Optical design has made a supplementary contribution to solving the issue of spherical aberration. If multiple lens elements of low refractive power, i.e. long focal lengths, are combined to a lens of high refractive power, spherical aberration is less of an issue than for a single lens element with a small focal length [30].

4. New techniques affecting optical design and optical material development

In the last half century, a couple of new techniques affecting conventional optical design and optical materials have been introduced: aspheres, diffractive optical elements, and opto-ceramics. Below, it will be briefly discussed if and how they will affect the general picture presented here. A comprehensive presentation about optical materials in the future is not given here, but can be found in [31]. Novel optical techniques like lightguiding or negative index materials are beyond the scope of this paper.

4.1 Aspheres

The development of production methods for aspherical surfaces supports the suppression of spherical aberration. However, this will not necessarily simplify the issue of glass selection. Instead, optical designers may choose the way to leave out some lens elements from their design and take the best available glass for the remaining elements. This approach is particularly useful for aspheres manufactured with the sophisticated process of magnetorheological finishing (MRF, [32]), Fig. 9.

 

Fig. 9. Asphere from SCHOTT AG.

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Precision moulding has been utilized as the less expensive alternative to the MRF process of asphere manufacturing. For precision moulding, glass selection has been a particular issue. To allow moulding at moderate temperatures, specifically to avoid an early wearing away of the moulds, special glasses with low glass transition temperatures had to be developed [2,33], Fig. 1.

4.2 Diffractive optical elements

Microstructuring techniques such as photolithography have enabled the manufacture of optical elements where the imaging is done by diffraction rather than by refraction. As diffraction is strongest at longer wavelengths, for diffractive optics, the sequence of foci is opposite to that of refractive lenses: f_red < f_yellow < f_blue [30]. This offers a supplementary way of colour correction. In praxi, however, the effect may be less than originally expected [34]. Never­theless, this approach is worth exploiting, in particular with respect to system miniaturization. Material selection is again an issue (high index materials are preferred; see [35]).

4.3 Opto-ceramics

Opto-ceramics are an interesting extension of the optical material toolbox. The different processing routes allow for the achievement of, e.g., very high packing densities and thus some extraordinary optical Abbe diagram positions with respect to refractive index and dispersion [5]. Of course, Kramers-Kronig applies to them as well as to any other material; therefore, this extension of the range of optical positions is limited. For example: [36] reports the fabrication of optically transparent polycrystalline ZrO₂ ceramics by a solid-state sintering process involving, first, vacuum sintering, and, second, hot isostatic pressing. For the composition 64.2mol% ZrO₂, 26.6mol% Y₂O₃, and 9.2mol% TiO₂ a refractive index n_d = 2.11 and an Abbe number ν_d = 39 were measured. The resultant material has a remarkable position in the Abbe diagram, one that is even beyond the scale of Fig. 1. In-line transmission (total transmission minus scatter fraction) at the Fraun­hofer d-line was measured at 68%, 10% below the theoretical limit. One half of these 10% was found to be due to scattering. For optical ceramics, scattering is the critical issue for use in optical imaging applications.

5. Conclusion

Optical imaging has a 400 year history now, from a start with telescopes and microscopes in the 17^th century to augmented reality in the 21^st century. As the limits to optical designers and material developers are not purely technical ones, but essentially caused by the fundamental laws of solid state physics (Kramers-Kronig relations), optical imaging is not an issue that may be solved once and for all. It is a permanent challenge that brings scientists back again and again to the same issues that have been worked on by ingenious pioneers like Otto Schott.