Optical element with a stack of layer packets, and method for producing the optical element

State of the art

0001The invention relates to an optical element with a stack of layer packages and a method for producing the optical element.

0002Well-known optical elements with interferometric anti-reflection coating, such as those from<patcit id="pcit0001" dnum="WO2016110339A1"><text>WO 2016/110339 A1</text></patcit> known, usually have a light reflectance of approximately 1%, calculated according to the standard DIN EN ISO 13666:2013-10. The color of the remaining residual reflection shows a strong variation when the viewing angle is changed. The variation extends practically over the entire visual color gamut.

disclosure of the invention

0003The object of the invention is to provide an optical element with an interferometric reflection-reducing layer system which has only a small viewing angle-dependent variation of the color of the residual reflection.

0004The objects are achieved by the features of the independent claims. Favorable embodiments and advantages of the invention emerge from the further claims, the description and the drawing.

0005Unless otherwise specified, the terms used in this disclosure should be understood in the sense of the standard DIN EN ISO 13666:2013-10 (EN ISO 13666:2012 (D/E)) and DIN EN ISO 11664-4:2012-06 (EN ISO 11664-4:2011) of the German Institute for Standardization eV.

0006According to section 4.2 of the standard DIN EN ISO 13666:2013-10, the term visible light, visible radiation or a visible wavelength range refers to optical radiation that can directly cause a light sensation in humans. Visible radiation generally refers to a wavelength range from 400 nm to 780 nm.

0007In the context of this disclosure, visible radiation can preferably refer to a wavelength range of 400 nm or 460 nm to 700 nm, corresponding to the maximum sensitivity of the human eye. This can simultaneously increase the design flexibility for the design of the filter properties and edge steepness.

0008According to section 15.1 of the standard DIN EN ISO 13666:2013-10, the term spectral reflectance, reflectance or reflectivity refers to the ratio of the spectral radiant power reflected by the respective material, surface or coating to the incident radiant power for a specific wavelength (λ). In the present case, the reflectivity refers to the reflectivity of the entire coating with its several high and low refractive index sublayers and not to the reflectivity of a single sublayer.

0009The invention is based on an optical element comprising a substrate and an interferometrically reflection-reducing layer system on at least one surface of the substrate, wherein the layer system comprises a stack of at least four successive layer packages, wherein each layer package comprises a first partial layer with a first optical thickness and a second partial layer with a second optical thickness different from the first optical thickness, wherein a refractive index of the respective first sublayer closer to the substrate is greater than a refractive index of the respective second sublayer of the stack further away from the substrate, wherein the layer system has a brightness, a chroma and a hue angle of a residual reflection color.

0010According to a favorable embodiment, it is proposed that the amount of a, in particular absolute or relative, change in the hue angle of the residual reflex color in an interval of a viewing angle with the limit values ​​0° and 30° relative to a surface normal to the layer system is smaller than the amount of a, in particular absolute or relative, change in the chroma in the interval of the viewing angle.

0011According to a favorable design, the scotopic reflectance in the interval of the viewing angle with the limit values ​​0° and 30° is at most 1.5%.

0012The chroma can also be referred to as color saturation. The hue angle can also be referred to as color angle.

0013By varying the layer thicknesses of the sublayers, a color-stable layer system can be advantageously provided, the residual reflex color of which does not change or only changes slightly even with a large change in the viewing angle. A color-stable residual reflex color can advantageously be achieved by a suitable combination of chroma and hue angle over a large viewing angle range.

0014The first sublayers of the layer packages in the stack that are closer to the substrate can be formed from the same first material.

0015The second partial layers further away from the substrate can also be made of an identical second material that is different from the first material of the first partial layers. In this case, it can be provided that in the layer package furthest from the substrate, between the first and second partial layers, a functional layer made of a third material is arranged, which has comparable refractive properties to the second partial layer. For calculation purposes, the functional layer can be assigned to the second sublayer if necessary. Alternatively, the materials of the first sublayers can vary in the stack. Alternatively, it can also be provided that the materials from which the second sublayers are formed vary in the stack.

0016The layer system can advantageously have four or five layer packages. Preferably, five layer packages are provided. More than five layer packages can also be provided.

0017According to a favorable design of the optical element, the scotopic reflectance Rv` in the interval of the viewing angle AOI with the limit values ​​0° and 30° can be at most 1.2%.

0018According to a favorable design of the optical element, in particular for a residual reflection color blue, the scotopic reflection coefficient Rv' at a viewing angle AOI of 0° can be approximately 0.7% and for viewing angles AOI between 0° and 30° between 0.6% and 0.7% and increase to a value of 1% for viewing angles AOI from 30° to 45°, and/or for a viewing angle AOI of 0° the hue angle can be approximately h=272°.

0019According to a favorable design of the optical element, in particular for a residual reflection color green, the scotopic reflectance Rv' at a viewing angle AOI of 0° can be about 0.6% and for viewing angles AOI between 0° and 30° between 0.6% and 0.7% and for viewing angles AOI from 30° to 45° increase to a value of 1.5%, and/or for a viewing angle AOI of 0° the hue angle can be about h=147°.

0020According to a favorable design of the optical element, in particular for a residual reflection color yellow, the scotopic reflection coefficient Rv' at a viewing angle AOI of 0° can be approximately 0.6% and for viewing angles AOI between 0° and 20° between 0.6% and 0.7% and for viewing angles AOI from 30° to 45° increase to a value of 2%, and/or for a viewing angle AOI of 0° the hue angle can be approximately h=84°.

0021According to a favorable design of the optical element, in particular for a residual reflex color red, the scotopic reflection coefficient Rv' can be about 0.35% at a viewing angle AOI of 0° and remain almost constant for viewing angles AOI between 0° and 20° and increase to a value of 1% for viewing angles AOI from 30° to 45°, and/or for a viewing angle AOI of 0° the hue angle can be about h=10°.

0022According to a favorable design of the optical element, a functional layer can be arranged between the first partial layer and the second partial layer in the layer package furthest from the substrate. The functional layer can advantageously act, for example, to increase electrical conductivity, to equalize mechanical stress or as a diffusion barrier.

0023According to a favorable design of the optical element, the chroma at the upper limit of the viewing angle can have a value of at most 16. Alternatively or additionally, the maximum value of the chroma in the interval of the viewing angle can be at most 16. This enables the realization of all reflex colors with high color constancy for a residual reflection, not only at the edge of the hue angle.

0024With a favorable design of the optical element, the hue angle in the interval of the viewing angle with the limit values ​​0° and 30° can change by a maximum of 15°, preferably by a maximum of 10°. The color impression of the residual reflection of the optical system remains completely or almost unchanged for an observer over a large range of the viewing angle.

0025According to a favorable design of the optical element, the amount of change in the hue angle in a second interval of a viewing angle from 0° to a limiting viewing angle with upper limit values ​​of 30° to 45° relative to the surface normal to the layer system can be smaller than the amount of change in the chroma in the second interval of the viewing angle and the amount of the chroma at the limiting viewing angle can be at least equal to 2. In particular, the hue angle h in the second interval can change by a maximum of 20°, preferably by a maximum of 15°. This advantageously results in a color-stable residual reflex color even with a larger variation in the viewing angle.

0026According to a favorable design of the optical element, the photopic reflectance in the interval of the viewing angle with the limit values ​​0° and 30° can be at most 1.5%, preferably at most 1.2%.

0027According to a favorable design of the optical element, the first partial layers can be formed from a highly refractive material.

0028Advantageously, the first sublayers may consist of at least one or more of the compounds Ta₂O₅, TiO₂, ZrO₂, Al₂O₃, Nd₂O₅, Pr₂O₃, PrTiOs, La₂O₃, Nb₂O₅, Y₂O₃, HfO₂, InSn oxide, Si₃N₄, MgO, CeO₂, ZnS and/or their modifications, in particular their other oxidation states.

0029If two or more compounds are contained in a first partial layer, they can be applied layer by layer or mixed in one layer, for example by simultaneous application.

0030These materials are known as materials with a high classical refractive index for use in optical elements, such as for coating spectacle lenses. However, the higher refractive index sublayers can also be SiO₂ or other low-refractive materials, as long as the refractive index of the entire sublayer is greater than 1.6, preferably at least 1.7, particularly preferably at least 1.8, most preferably at least 1.9.

0031According to a favorable design of the optical element, the second partial layers can be formed from a low-refractive material.

0032The lower refractive index sublayers can contain at least one of the materials MgF₂, SiO, SiO₂, SiO₂ with additions of Al, silanes, siloxanes in pure form or with their fluorinated derivatives. However, the lower refractive index partial layers can also contain a mixture of SiO₂ and Al₂O₃ Preferably, the lower refractive index sublayers can contain at least 80 percent by weight of SiO₂, particularly preferably at least 90 percent by weight SiO₂ contain.

0033Preferably, the refractive index of the low-refractive index sublayers is at most 1.55, preferably at most 1.48, particularly preferably at most 1.4. These refractive indices refer to normal conditions at a temperature of 25° C and a reference wavelength of the light intensity used of 550 nm.

0034Typical examples of layered materials with different refractive indices are silicon dioxide (SiO₂) with a refractive index of 1.46, aluminum oxide (Al₂O₃) with a refractive index of 1.7, zirconium dioxide (ZrO₂) with a refractive index of 2.05, praseodymium titanium oxide (PrTiOs) with a refractive index of 2.1, titanium oxide (TiO₂) and zinc sulfide (ZnS), each with a refractive index of 2.3. These values ​​represent average values ​​that can vary by up to 10% depending on the coating process and layer thickness.

0035Common optical glasses have refractive indices between 1.5 and 2.0. Layered materials with refractive indices less than 1.5 such as MgF₂, SiO₂, Al₂O₃ are therefore referred to as low-refractive materials in combination with optical glasses, layer materials with refractive indices greater than 2.0 such as ZrO₂, PrTiOs, TiO₂, ZnS in combination with optical glasses are referred to as highly refractive materials.

0036The difference in the refractive indices between the high-refractive and low-refractive materials of the first and second sublayers is preferably at least 0.2 to at least 0.5, depending on the coating process and layer thickness.

0037The materials used for this type of coating are the typical materials that are applied to a substrate in optics using, for example, PVD processes (PVD = Physical Vapour Deposition) or CVD processes (CVD = Chemical Vapour Deposition).

0038According to a favorable embodiment of the optical element, at least the first partial layers can be formed from a same first material and the second partial layers can be formed at least predominantly from a same second material.

0039Optionally, the second sublayers can be made of the same second material and only have a functional layer between the first sublayer and the second sublayer in the layer package furthest from the substrate. The functional layer can have a low refractive index and, if necessary, can be added to the second sublayer for calculation purposes.

0040A design of an optical element according to the invention can be carried out using a favorable method.

0041The optical element comprises a substrate and an interferometrically reflection-reducing layer system on at least one surface of the substrate, wherein the layer system comprises a stack of at least four successive layer packages, wherein each layer package comprises a first partial layer with a first optical thickness and a second partial layer with a second optical thickness different from the first optical thickness, wherein a refractive index of the respective first sublayer closer to the substrate is greater than a refractive index of the respective second sublayer of the stack further away from the substrate, wherein the layer system has a brightness, a chroma and a hue angle of a residual reflection color. The amount of a, in particular absolute or relative, change in the hue angle of the residual reflex color in an interval of a viewing angle with the limit values ​​0° and 30° relative to a surface normal to the layer system is preferably smaller than the amount of a, in particular absolute or relative, change in the chroma in the interval of the viewing angle.

0042In the method, the following steps can be carried out: defining a layer design, comprising at least a first material for high-refractive index sublayers and a second material for low-refractive index sublayers, number of desired layer packages with the sublayers, starting values ​​of the thickness of the sublayers; Defining target color values, comprising brightness, chroma and hue angle at least at limit values ​​for an interval of a viewing angle with limit values ​​of 0° and 30°; and performing an optimization process for varying the individual layer thicknesses until an optimization goal is reached.

0043Conveniently, a value of 16 or less can be selected for the chroma at the upper limit of the viewing angle. Alternatively or additionally, a maximum value of 16 or less can be selected for the chroma in the viewing angle interval. This enables the realization of all reflex colors with high color constancy for a residual reflection, not only at the edge of the hue angle.

0044Conveniently, the target color values ​​at the limits of the interval can be chosen to be the same or similar.

0045In particular, maximum deviations for the hue angles of different residual reflex colors can be specified.

0046For the residual reflex color blue, a favorable permissible change Δh of the hue angle in the viewing angle interval from 0° to 30° can preferably be at most Δh=4°, particularly preferably at most Δh=3.5°. A favorable permissible change Δh of the hue angle in the viewing angle interval from 0° to 33° can preferably be at most Δh=5°, particularly preferably at most Δh=4.5°.

0047For a residual reflex color green, a favorable permissible change Δh of the hue angle in the viewing angle interval from 0° to 30° can preferably be at most Δh=3°, particularly preferably at most Δh=2°. A favorable permissible change Δh of the hue angle in the viewing angle interval from 0° to 45° can preferably be at most Δh=5°, particularly preferably at most Δh=4.6°.

0048For a residual reflex color yellow, a favorable permissible change Δh of the hue angle in the viewing angle interval from 0° to 30° can preferably be at most Δh=1.5°, particularly preferably at most Δh=0.9°. A favorable permissible change Δh of the hue angle in the viewing angle interval from 0° to 45° can preferably be at most Δh=5°, particularly preferably at most Δh=4.6°.

0049For a residual reflex color red, a favorable permissible change Δh of the hue angle in the viewing angle interval from 0° to 30° can preferably be at most Δh=6°, particularly preferably at most Δh=5.3°. A favorable permissible change Δh of the hue angle in the viewing angle interval from 0° to 45° can preferably be at most Δh=20°, particularly preferably at most Δh=16.8°.

drawing

0050Further advantages emerge from the following description of the drawings. The figures show embodiments of the invention. The figures, the description and the claims contain numerous features in combination. The person skilled in the art will also expediently consider the features individually and combine them into further useful combinations.

Examples include:

0051<dl id="dl0001" compact="compact"><dt>Fig. 1</dt><dd>an embodiment of the invention in which a layer system has four layer packages on a substrate;</dd><dt>Fig. 2</dt><dd>an embodiment of the invention in which a layer system has five layer packages on a substrate;</dd><dt>Fig. 3</dt><dd>a reflectivity of a layer system according to the invention with a residual reflex color blue in the wavelength range between 280 nm and 1400 nm;</dd><dt>Fig. 4</dt><dd>a detail of the representation in<figref idref="f0003">Figure 3</figref> in the wavelength range 380 nm to 780 nm;</dd><dt>Fig. 5</dt><dd>Depending on the angle of incidence between 0° and 45°, a gradient of the chroma C* with the residual reflex color blue according to<figref idref="f0003">Figure 3</figref>;</dd><dt>Fig. 6</dt><dd>Depending on the angle of incidence between 0° and 45°, a gradient of the brightness L* of the residual reflex colour blue according to<figref idref="f0003">Figure 3</figref>;</dd><dt>Fig. 7</dt><dd>Depending on the angle of incidence between 0° and 45°, a gradient of the hue angle h with the residual reflex color blue according to<figref idref="f0003">Figure 3</figref>;</dd><dt>Fig. 8</dt><dd>depending on the angle of incidence between 0° and 45° a progression of the photopic reflectance Rv and the scotopic reflectance Rv` with the residual reflex colour blue according to<figref idref="f0003">Figure 3</figref>;</dd><dt>Fig. 9</dt><dd>a reflectivity of a layer system according to the invention with a residual reflex color green in the wavelength range between 280 nm and 1400 nm;</dd><dt>Fig. 10</dt><dd>a detail of the representation in<figref idref="f0006">Figure 9</figref> in the wavelength range 380 nm to 780 nm;</dd><dt>Fig. 11</dt><dd>Depending on the angle of incidence between 0° and 45°, a gradient of the chroma C* with the residual reflex color green to<figref idref="f0006">Figure 9</figref>;</dd><dt>Fig. 12</dt><dd>Depending on the angle of incidence between 0° and 45°, a gradient of the brightness L* of the residual reflex color green to<figref idref="f0006">Figure 9</figref>;</dd><dt>Fig. 13</dt><dd>Depending on the angle of incidence between 0° and 45°, a gradient of the hue angle h with the residual reflex color green to<figref idref="f0006">Figure 9</figref>;</dd><dt>Fig. 14</dt><dd>depending on the angle of incidence between 0° and 45° a progression of the photopic reflectance Rv and the scotopic reflectance Rv` with the residual reflex colour green according to<figref idref="f0006">Figure 9</figref>;</dd><dt>Fig. 15</dt><dd>a reflectivity of a layer system according to the invention with a residual reflex color yellow in the wavelength range between 280 nm and 1400 nm;</dd><dt>Fig. 16</dt><dd>a detail of the representation in<figref idref="f0009">Figure 15</figref> in the wavelength range 380 nm to 780 nm;</dd><dt>Fig. 17</dt><dd>Depending on the angle of incidence between 0° and 45°, a gradient of the chroma C* with the residual reflex color yellow to<figref idref="f0009">Figure 15</figref>;</dd><dt>Fig. 18</dt><dd>Depending on the angle of incidence between 0° and 45°, the brightness L* of the residual reflex colour yellow varies according to<figref idref="f0009">Figure 15</figref>;</dd><dt>Fig. 19</dt><dd>Depending on the angle of incidence between 0° and 45°, a gradient of the hue angle h with the residual reflex color yellow to<figref idref="f0009">Figure 15</figref>;</dd><dt>Fig. 20</dt><dd>depending on the angle of incidence between 0° and 45° a progression of the photopic reflectance Rv and the scotopic reflectance Rv` with the residual reflex colour yellow according to<figref idref="f0009">Figure 15</figref>;</dd><dt>Fig. 21</dt><dd>a reflectivity of a layer system according to the invention with a residual reflex color red in the wavelength range between 280 nm and 1400 nm,</dd><dt>Fig. 22</dt><dd>a detail of the representation in<figref idref="f0012">Figure 21</figref> in the wavelength range 380 nm to 780 nm;</dd><dt>Fig. 23</dt><dd>Depending on the angle of incidence between 0° and 45°, a gradient of the chroma C* with the residual reflex color red according to<figref idref="f0012">Figure 21</figref>;</dd><dt>Fig. 24</dt><dd>Depending on the angle of incidence between 0° and 45°, a gradient of the brightness L* of the residual reflex colour red according to<figref idref="f0012">Figure 21</figref>;</dd><dt>Fig. 25</dt><dd>Depending on the angle of incidence between 0° and 45°, a gradient of the hue angle h with the residual reflex color red to<figref idref="f0012">Figure 21</figref>;</dd><dt>Fig. 26</dt><dd>depending on the angle of incidence between 0° and 45° a progression of the photopic reflectance Rv and the scotopic reflectance Rv` with the residual reflex colour red according to<figref idref="f0012">Figure 21</figref>;</dd><dt>Fig. 27</dt><dd>a polar diagram with measured values ​​on layer systems according to the invention with different residual reflex colors and a layer system according to the prior art in a first interval of a viewing angle of 0° to 30°;</dd><dt>Fig. 28</dt><dd>the polar diagram according to<figref idref="f0015">Figure 27</figref> in a further interval of viewing angle from 0° to 45°;</dd><dt>Fig. 29</dt><dd>a favorable method for designing a layer system according to the invention.</dd></dl>

embodiments of the invention

0052In the figures, components of the same type or with the same effect are numbered with the same reference symbols. The figures show only examples and are not to be understood as limiting.

0053Directional terminology used below with terms such as "left", "right", "top", "bottom", "before", "behind", "after" and the like is intended only to facilitate understanding of the figures and is in no way intended to limit the scope of the invention. The components and elements shown, their design and use may vary in accordance with the considerations of a person skilled in the art and may be adapted to the respective applications.

0054The<figref idref="f0001">Figures 1</figref> and<figref idref="f0002">2</figref> show an example of an optical element 100, for example a spectacle lens, according to an embodiment of the invention.

0055The optical element 100 comprises<figref idref="f0001">Figure 1</figref> a layer system 10 with a stack 40 of four layer packages 42, 44, 46, 48 on a surface 22 of a substrate 20.

0056In<figref idref="f0002">Figure 2</figref> the optical element 100 comprises a layer system 10 with a stack 40 of five layer packages 42, 44, 46, 48, 50 on a surface 22 of a substrate 20.

0057Except for the different number of shift packages 42, 44, 46, 48, 50 (four in<figref idref="f0001">Figure 1</figref> and five in<figref idref="f0002">Figure 2</figref>) the further statements of a general and specific nature apply to<figref idref="f0001">Figure 1</figref> also for the design in<figref idref="f0002">Figure 2</figref>unless otherwise stated.

0058The layer system is viewed by an observer at a viewing angle AOI of 0° up to a critical angle, for example 30°, measured from the surface normal 70.

0059The substrate 20 is, for example, a plastic, in particular a transparent plastic for a spectacle lens.

0060In the context of the present disclosure, the term spectacle lens refers in particular to a coated spectacle lens according to section 8.1.13 of the standard DIN EN ISO 13666:2013-10, i.e. a spectacle lens to which one or more surface coatings have been applied, in particular to change one or more of its properties.

0061Preferably, such lenses can be used particularly advantageously as glasses (with and without correction), sunglasses, ski goggles, workplace goggles, and glasses in conjunction with head-mounted displays.

0062In the context of the present disclosure, the term spectacle lens can also include spectacle lens semi-finished products, in particular a spectacle lens blank or spectacle lens semi-finished product according to section 8.4.2 of the standard DIN EN ISO 13666:2013-10, i.e. a lens blank or blank with only one optically finished surface.

0063With regard to the arrangements in the<figref idref="f0001">Figures 1</figref> and<figref idref="f0002">2</figref> the opposite surface 24 of the substrate 20 can optionally have another, similar or identical layer system 10, no coating or only a protective coating.

0064As the lowest layer on the substrate 20, the layer system 10 usually has a single-layer or multi-layer intermediate layer 32, for example to improve the adhesion of the stack 40 and/or as scratch protection for the substrate 20. This intermediate layer 32 can consist, for example, of substoichiometric low-refractive metal oxides, chromium, silanes or siloxanes. The intermediate layer 32 is not relevant for further considerations.

0065On the intermediate layer 32 are<figref idref="f0001">Figure 1</figref> the four layer packages 42, 44, 46, 48 of the stack 40 are arranged one after the other, wherein each of the layer packages 42, 44, 46, 48 consists of a sub-layer 60 closer to the substrate followed by a sub-layer 62 further away from the substrate.

0066Preferably, each of the sublayers 60 closer to the substrate is formed from an identical first material. Preferably, the first material is a higher refractive index material with a first refractive index n1.

0067Preferably, each of the sublayers 62 remote from the substrate is formed from an identical second material. Preferably, the second material is a low-refractive material with a second refractive index n2. The refractive index n1 is greater than the refractive index n2, preferably the difference between the refractive indices n1, n2 is at least 0.2, preferably up to at least 0.5.

0068The order of the sublayers 60, 62 remains the same in the stack 40, so that in each layer package 42, 44, 46, 48 the respective sublayer 60 closer to the substrate is always the one with the higher refractive index and the respective sublayer 62 further away from the substrate is always the one with the lower refractive index of the sublayers 60, 62.

0069In particular, the higher refractive index sublayers 60 can be layers made of high refractive index materials and the lower refractive index sublayers 62 can be layers made of low refractive index materials.

0070The layer packages 42, 44, 46, 48 in the stack 40 differ only in their respective thickness and/or in the thicknesses of the individual sub-layers 60, 62 in the respective layer package 42, 44, 46, 48.

0071The stack 40 is closed in a manner known per se with a cover layer 34, which serves, for example, to maintain the layer system 10. The cover layer 34 is applied to the last optically relevant partial layer 62 of the uppermost layer package 48 of the stack 40 and can contain fluorine-containing molecules. The cover layer 34 typically provides the stack 40 with improved care properties, with properties such as a water-repellent and oil-repellent function with a surface energy of typically less than 15 mN/m.

0072The cover layer 34 is not relevant for further considerations.

0073The uppermost layer package 48 furthest from the substrate (or layer package 50 in<figref idref="f0002">Figure 2</figref>) optionally has a functional layer 64 between the sub-layer 60 closer to the substrate and the sub-layer 62 further away from the substrate, which can act, for example, to increase the electrical conductivity, to equalize mechanical stress or as a diffusion barrier. This functional layer 64 can be made of a low-refractive material, and can be alloyed with other metal oxides such as aluminum. For calculation and simulation purposes of the optical properties, the functional layer 64 of the lower refractive index sublayer 62 of the uppermost layer package 48 furthest from the substrate (or layer package 50 in<figref idref="f0002">Figure 2</figref>) or, where appropriate, may be disregarded, for example in the case of a relatively small layer thickness.

0074The optical properties of the stack 40 of the layer system 10 can be simulated mathematically using known calculation methods and/or optimization methods. The layer system 10 is then produced with the determined layer thicknesses of the individual partial layers 60, 62 of the layer packages 42, 44, 46, 48.

0075During the production of optical layer systems 10, the optical properties of the layer system 10 are adjusted during the production of the partial layers 60, 62. For example, the<patcit id="pcit0002" dnum="WO2016110339A1"><text>WO 2016/110339 A1</text></patcit> A known process can be used, which is briefly outlined below. Using the known process, various optical effects such as mirroring or reflection reduction can be achieved in a material system by only changing the layer thicknesses, but the material used remains the same. However, other processes are also possible.

0076Through a<patcit id="pcit0003" dnum="WO2016110339A1"><text>WO 2016/110339 A1</text></patcit> By varying the layer package thicknesses described above with the same materials, different reflectivities can be achieved, especially for a reflection-reducing effect. This is achieved by minimizing or optimizing a parameter σ. The parameter σ is in turn a function of the layer thicknesses of the partial layers 60, 62, or of ratios of the optical thicknesses t1, t2 of the partial layers 60, 62 of each of the four layer packages 42, 44, 46, 48 according to<figref idref="f0001">Figure 1</figref> or five shift packages 42, 44, 46, 48, 50 after<figref idref="f0002">Figure 2</figref> in stack 40.

0077At a certain wavelength λ, the optical thickness t of a layer, also called FWOT (full wave optical thickness), is determined as<maths id="math0001"><math display="block"><mi mathvariant="normal">t</mi><mo>=</mo><mfrac><mi mathvariant="normal">d</mi><mi mathvariant="normal">λ</mi></mfrac><mo>⋅</mo><mi mathvariant="normal">n</mi></math><img file="EP4502702A2_D0001.tif" /></maths> where d is the layer thickness, λ is the design wavelength and n is the refractive index of the sublayer 60, 62.

0078A reflection-reducing effect by the stack 40 can be achieved for a predeterminable reflectivity R_m of the stack 40 can be achieved if the product of reflectivity R_m and the parameter σ is set to less than 1:<maths id="math0002"><math display="block"><msub><mi mathvariant="normal">R</mi><mi mathvariant="normal">m</mi></msub><mo>⋅</mo><mi mathvariant="normal">σ</mi><mo>&lt;</mo><mn>1</mn></math><img file="EP4502702A2_D0002.tif" /></maths>

0079The reflectivity R_m, also called reflection factor, describes the ratio of reflected to incident intensity of a light beam as an energy quantity. The reflectivity R_m is conveniently averaged over the range of light from 380 nm to 800 nm and related to 100%.

0080Such a condition R_m·σ&lt;1 can be used as a boundary condition for an optimization process of the method for producing the layer system 10.

0081The optical thicknesses t1, t2 of the first and second partial layers 60, 62 of the layer packages 42, 44, 46, 48 are determined by determining the parameter σ by means of an optimization method, preferably by means of variational calculus.

0082Preferably, the thicknesses of the respective sub-layers 60, 62 in four layer packages 42, 44, 46, 48 in the stack 40 are dependent on a quotient v_i (with i=1, 2, 3, 4) of the first optical thickness t1 of the higher refractive first partial layer 60 and the second optical thickness t2 of the lower refractive second partial layer 62 of the respective layer package 42, 44, 46, 48.

0083The indices i=1, 2, 3, 4 represent the order of the layer packages 42, 44, 46, 48 on the substrate 20. Accordingly, v₁ for the substrate next layer package 42 and v₄ for the layer package furthest from the substrate 48.

0084For a stack of four consecutive layer packages 42, 44, 46, 48, the parameter σ can be calculated from<maths id="math0003"><math display="block"><mi mathvariant="normal">σ</mi><mo>=</mo><mfrac><mrow><msub><mi mathvariant="normal">v</mi><mn>1</mn></msub><mo>+</mo><msub><mi mathvariant="normal">v</mi><mn>2</mn></msub></mrow><mrow><msub><mi mathvariant="normal">v</mi><mn>3</mn></msub><mo>+</mo><msub><mi mathvariant="normal">v</mi><mn>4</mn></msub></mrow></mfrac></math><img file="EP4502702A2_D0003.tif" /></maths> be determined.

0085The first and second sublayers 60, 62 are produced with the parameters calculated in this way, in particular the optical thicknesses t1, t2 of the sublayers 60, 62 of the stack 40.

0086In an advantageous embodiment, in a layer system 10 according to<figref idref="f0002">Figure 2</figref> the parameter σ for a stack 40 with five consecutive layer packages 42, 44, 46, 48, 50 from the relationship<maths id="math0004"><math display="block"><mi mathvariant="normal">σ</mi><mo>=</mo><mfrac><msub><mi mathvariant="normal">v</mi><mn>1</mn></msub><mstyle displaystyle="true"><msubsup><mo>∑</mo><mrow><mi mathvariant="normal">i</mi><mo>=</mo><mn>2</mn></mrow><mi>nmax</mi></msubsup><msub><mi mathvariant="normal">v</mi><mi mathvariant="normal">i</mi></msub></mstyle></mfrac></math><img file="EP4502702A2_D0004.tif" /></maths> where i= runs from 2 to nmax=5.

0087The indices i=1, 2, 3, 4, 5 stand for the order of the layer packages 42, 44, 46, 48, 50 on the substrate 20. Accordingly, v₁ for the substrate next layer package 42 and v₅ for the layer package furthest from the substrate 50.

0088It is known to specify perceptual colors in the so-called CIE-L*a*b* color space (simplified CIELab color space) in Cartesian coordinates, as set out in DIN EN ISO 11664-4:2012-06 (EN ISO 11664-4:2011).

0089L* is the CIELab brightness, a*, b* are the CIELab coordinates, C* is the CIELab chroma and h_away the CIELab hue angle.

0090The L* axis describes the brightness (luminance) of the color with values ​​from 0 to 100. The L* axis is perpendicular to the a*b* plane at the zero point. It can also be called the neutral gray axis, since all achromatic colors (gray tones) are contained between the end points black (L*=0) and white (L*=100).

0091Green and red are opposite each other on the a* axis, the b* axis runs between blue and yellow. Complementary colors are opposite each other by 180°, and in their middle, ie the coordinate origin a*=0, b*=0, is gray.

0092The a* axis describes the green or red component of a color, with negative values ​​representing green and positive values ​​representing red. The b* axis describes the blue or yellow component of a color, with negative values ​​representing blue and positive values ​​representing yellow.

0093The a* values ​​range from approximately -170 to +100, the b* values ​​from -100 to +150, whereby the maximum values ​​are only reached at medium brightness of certain colors. The CIELab color solid has its greatest extent in the medium brightness range, but this varies in height and size depending on the color range.

0094The CIELab hue angle h_away must be between 0° and 90° if both a* and b* are positive, between 90° and 180° if b* is positive and a* is negative, between 180° and 270° if both a* and b* are negative, and between 270° and 360° if b* is negative and a* is positive.

0095In the CIE L*C*h color space (simplified CIELCh color space), the Cartesian coordinates of the CIELab color space are transformed into polar coordinates. The cylindrical coordinates C* (chroma, relative color saturation, distance from the L axis in the center) and h (hue angle, angle of the hue in the CIELab color circle) are specified. The CIELab brightness L* remains unchanged.

0096The hue angle h results from the a* and b* axes<maths id="math0005"><math display="block"><mi mathvariant="normal">h</mi><mo>=</mo><mi>arctan</mi><mfenced><mfrac><mrow><mi mathvariant="normal">b</mi><mo>*</mo></mrow><mrow><mi mathvariant="normal">a</mi><mo>*</mo></mrow></mfrac></mfenced></math><img file="EP4502702A2_D0005.tif" /></maths>

0097The hue angle h represents the color of the residual reflection of the anti-reflective layer system 10.

0098The chroma C* results in<maths id="math0006"><math display="block"><mi mathvariant="normal">C</mi><mo>*</mo><mo>=</mo><msqrt><mrow><msup><mfenced><mi mathvariant="normal">a</mi><mo>*</mo></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mi mathvariant="normal">b</mi><mo>*</mo></mfenced><mn>2</mn></msup></mrow></msqrt></math><img file="EP4502702A2_D0006.tif" /></maths>

0099The chroma C* is also called color depth.

0100In known anti-reflective coatings, the color of the residual reflection is optimized for a perpendicular incidence of light on the layer system 10 (AOI=0°) and varies greatly when the viewing angle AOI changes.

0101As shown in the following figures for various residual reflex colors blue, green, yellow, red, a layer system 10 according to the invention can be formed with four layer packages 42, 44, 46, 48 or five layer packages 42, 44, 46, 48, 50, which remains largely color-stable even when the viewing angle AOI varies in the range between 0° and 30°. In other words, the color impression of the residual reflection practically does not change when an observer views the optical element 100 at viewing angles AOI in this angular interval.

0102The<figref idref="f0003 f0004 f0005">Figures 3 to 8</figref> show values ​​of a layer system 10 according to the invention with the residual reflex color blue.

0103<figref idref="f0003">Figure 3</figref> shows a reflectivity R in percent of the layer system 10 according to the invention in the wavelength range between 280 nm and 1400 nm, and<figref idref="f0003">Figure 4</figref> shows a detail of the representation in<figref idref="f0003">Figure 3</figref> in the wavelength range 380 nm to 780 nm. The reflectivity R of the layer system 10 is in plan view of the layer system 10, ie at small angles AOI around 0° measured from the surface normal 70 (<figref idref="f0001">Figure 1</figref>, <figref idref="f0002">2</figref>), certainly.

0104<figref idref="f0004">Figure 5</figref> shows a gradient of the chroma C* depending on the viewing angle AOI between 0° and 45°,<figref idref="f0004">Figure 6</figref> shows a brightness curve L* depending on the viewing angle AOI between 0° and 45°,<figref idref="f0005">Figure 7</figref> shows a progression of the hue angle h between 0° and 45° depending on the viewing angle AOI.<figref idref="f0005">Figure 8</figref> shows a progression of the photopic reflectance Rv and the scotopic reflectance Rv' depending on the viewing angle AOI between 0° and 45°.

0105The reflectivity R drops from about 55% at 280 nm with a small maximum around 380 nm and slowly increases again at about 730 nm (<figref idref="f0003">Figure 3</figref>). As can be seen, the reflectivity R between 480 nm and 780 nm is very low and is below 1% (<figref idref="f0003">Figure 4</figref>). Between 530 nm and 730 nm, the reflectivity R is sometimes even below 0.5%.

0106In the AOI interval between 0° and 45°, the chroma C* falls from about 6.5 at 0° to about 2.5 at 30° to 0 at about 42°, and then increases (<figref idref="f0004">Figure 5</figref>). At 45°, the chroma value C* is approximately 1. The highest chroma value C* is in the AOI interval for the color blue at C*=7. The highest chroma value C* is at the lower limit of the AOI interval.

0107In the AOI interval between 0° and 25°, the brightness L* varies little with a slight decrease and increases at 30° from about L*=4.5 to about L*=8 at 45° (<figref idref="f0004">Figure 6</figref>).

0108In the AOI interval between 0° and 45°, the hue angle h drops slightly between 0° and 30° from about h=272° to just over h=268° and then drops more steeply to h=260° at AOI=40° (<figref idref="f0005">Figure 7</figref>).

0109For the residual reflex color blue, the change Δh in the hue angle h between 0°≤AOI≤30° is preferably at most Δh=4°, particularly preferably at most Δh=3.5°. The change in the hue angle h between 0°≤AOI≤33° is preferably at most Δh=5°, particularly preferably at most Δh=4.5°.

0110In the AOI interval between 0° and 45°, the curves of the photopic reflectance Rv and the scotopic reflectance Rv` practically do not vary and remain at 0.5 (Rv) and 0.7-0.6 (Rv`), respectively, and increase between AOI=30° and 45° to a value of 1 (<figref idref="f0005">Figure 8</figref>).

0111The<figref idref="f0006 f0007 f0008">Figures 9 to 14</figref> show values ​​of a layer system 10 according to the invention with the residual reflex color green.

0112<figref idref="f0006">Figure 9</figref> shows a reflectivity R of the layer system 10 according to the invention in percent in the wavelength range between 280 nm and 1400 nm, and<figref idref="f0006">Figure 10</figref> shows a detail of the representation in<figref idref="f0006">Figure 9</figref> in the wavelength range 380 nm to 780 nm. The reflectivity R of the layer system 10 is in plan view of the layer system 10, ie at small angles AOI around 0° measured from the surface normal 70 (<figref idref="f0001">Figure 1</figref>, <figref idref="f0002">2</figref>), certainly.

0113<figref idref="f0007">Figure 11</figref> shows a gradient of the chroma C* depending on the viewing angle AOI between 0° and 45°,<figref idref="f0007">Figure 12</figref> shows a brightness curve L* depending on the viewing angle AOI between 0° and 45°,<figref idref="f0008">Figure 13</figref> shows a progression of the hue angle h between 0° and 45° depending on the viewing angle AOI.<figref idref="f0008">Figure 14</figref> shows a progression of the photopic reflectance Rv and the scotopic reflectance Rv' depending on the viewing angle AOI between 0° and 45°.

0114The reflectivity R drops from about 55% at 280 nm with a small maximum around 380 nm and slowly increases again at about 730 nm (<figref idref="f0006">Figure 9</figref>). As can be seen, the reflectivity R between 430 nm and 730 nm is very low and is less than 1% (<figref idref="f0007">Figure 11</figref>At around 530 nm and between about 600 nm and about 700 nm, the reflectivity R is sometimes even below 0.5%.

0115In the AOI interval between 0° and 45°, the chroma C* increases from about C*=5 at 0° to about C*=6 at 30°, continues to increase slightly and then decreases at 40°<figref idref="f0007">Figure 11</figref>). At 45°, the chroma value C* is approximately 5. The highest chroma value C* is in the AOI interval for the color green at C*=7. The highest chroma value C* is close to the upper limit of the AOI interval.

0116In the AOI interval between 0° and 30°, the brightness L* is almost constant with L*=5.5 and starts increasing at 25° and reaches L*=10 at 45° (<figref idref="f0007">Figure 12</figref>).

0117In the AOI interval between 0° and 45°, the hue angle h is almost constant between 0° and 20° with about h=147° and begins to increase slightly from 20° to just over h=155° at AOI=40° (<figref idref="f0008">Figure 13</figref>).

0118For the residual reflex color green, the change Δh in the hue angle h between 0°≤AOI≤30° is preferably at most Δh=5°, particularly preferably Δh=2°. The change Δh in the hue angle h between 0°≤AOI≤45° is preferably at most Δh=5°, particularly preferably at most Δh=4.6°.

0119In the AOI interval between 0° and 45°, the curves of the photopic reflectance Rv and the scotopic reflectance Rv` are practically equal and are almost constant between 0° and 30° and both remain at 0.6 (Rv) and 0.7-0.6 (Rv`), respectively. The curves increase between AOI=30° to 45° to a value of 1.5 (<figref idref="f0008">Figure 14</figref>).

0120The<figref idref="f0009 f0010 f0011">Figures 15 to 20</figref> show values ​​of a layer system 10 according to the invention with the residual reflex color yellow.

0121<figref idref="f0009">Figure 15</figref> shows a reflectivity R in percent of the layer system 10 according to the invention in the wavelength range between 280 nm and 1400 nm, and<figref idref="f0009">Figure 16</figref> shows a detail of the representation in<figref idref="f0006">Figure 9</figref> in the wavelength range 380 nm to 780 nm. The reflectivity R of the layer system 10 is in plan view of the layer system 10, ie at small angles AOI around 0° measured from the surface normal 70 (<figref idref="f0001">Figure 1</figref>, <figref idref="f0002">2</figref>), certainly.

0122<figref idref="f0010">Figure 17</figref> shows a gradient of the chroma C* depending on the viewing angle AOI between 0° and 45°,<figref idref="f0010">Figure 18</figref> shows a brightness curve L* depending on the viewing angle AOI between 0° and 45°,<figref idref="f0011">Figure 19</figref> shows a progression of the hue angle h between 0° and 45° depending on the viewing angle AOI.<figref idref="f0011">Figure 20</figref> shows a progression of the photopic reflectance Rv and the scotopic reflectance Rv' depending on the viewing angle AOI between 0° and 45°.

0123The reflectivity R drops from about 20% at 280 nm with a slightly higher maximum of just over 30% around 330 nm, then remains low and slowly increases again at about 680 nm (<figref idref="f0009">Figure 15</figref>). As can be seen, the reflectivity R between 430 nm and 580 nm is very low and is less than 1% (<figref idref="f0009">Figure 16</figref>).

0124In the AOI interval between 0° and 45°, the chroma C* increases slowly from about 6.5 at 0° to about 9 at 30°, has a maximum of 10 at about 35° and then decreases again (<figref idref="f0010">Figure 17</figref>). At 45°, the chroma value C* is 8. The highest chroma value C* in the AOI interval for the color yellow is C*=10 or just below. The highest chroma value C* is just below the upper limit of the AOI interval.

0125In the AOI interval between 0° and 25°, the brightness L* shows a slight increase of about L*=6.5 at 0° and L*=10 at about 35° and then slowly decreases again (<figref idref="f0010">Figure 18</figref>).

0126In the AOI interval between 0° and 45°, the hue angle h drops slightly between 0° and 30° from about h=84° to about h=83° and then drops more steeply from 35° (<figref idref="f0011">Figure 19</figref>).

0127For a residual reflex color yellow, the change Δh of the hue angle h between 0°≤AOI≤30° is preferably at most Δh=1.5°, particularly preferably Δh=0.9°. The change Δh of the hue angle h between 0°≤AOI≤45° is preferably at most Δh=5, particularly preferably at most Δh=4.6°.

0128In the AOI interval between 0° and 45°, the curves of the photopic reflectance Rv and the scotopic reflectance Rv' run parallel to each other and increase only slightly up to 20° and remain at 0.5 (Rv`) or 0.7-0.6 (Rv) and increase between AOI=30° and 45° to a value of 2 (<figref idref="f0011">Figure 20</figref>).

0129The<figref idref="f0012 f0013 f0014">Figures 21 to 26</figref> show values ​​of a layer system 10 according to the invention with the residual reflex color red.

0130<figref idref="f0012">Figure 21</figref> shows a reflectivity R in percent of the layer system 10 according to the invention in the wavelength range between 280 nm and 1400 nm, and<figref idref="f0012">Figure 22</figref> shows a detail of the representation in<figref idref="f0006">Figure 9</figref> in the wavelength range 380 nm to 780 nm. The reflectivity R of the layer system 10 is in plan view of the layer system 10, ie at small angles AOI around 0° measured from the surface normal 70 (<figref idref="f0001">Figure 1</figref>, <figref idref="f0002">2</figref>), certainly.

0131<figref idref="f0013">Figure 23</figref> shows a gradient of the chroma C* depending on the viewing angle AOI between 0° and 45°,<figref idref="f0013">Figure 24</figref> shows a brightness curve L* depending on the viewing angle AOI between 0° and 45°,<figref idref="f0014">Figure 25</figref> shows a progression of the hue angle h between 0° and 45° depending on the viewing angle AOI.<figref idref="f0014">Figure 26</figref> shows a progression of the photopic reflectance Rv and the scotopic reflectance Rv' depending on the viewing angle AOI between 0° and 45°.

0132The reflectivity R shows a maximum of about 20% at 330 nm and then drops to low values ​​and slowly increases again at about 530 nm (<figref idref="f0012">Figure 21</figref>). As can be seen, the reflectivity R between 430 nm and 600 nm is very low and is below 0.5% (<figref idref="f0012">Figure 22</figref>).

0133In the AOI interval between 0° and 45°, the chroma C* increases from about C*=4 at 0° to about C*=10 at 30°, and then remains constant between 35° and 45° (<figref idref="f0013">Figure 23</figref>). At 45°, the chroma value C* is at most 14, e.g. between 13 and 14. The highest chroma value C* is in the AOI interval for the color red at C*=13. The highest chroma value C* is close to the upper limit of the AOI interval.

0134In the AOI interval between 0° and 25°, the brightness L* varies little with a slight increase of about L*=3 and increases at 20° from about L*=4 to about L*=14 at 45° (<figref idref="f0013">Figure 24</figref>).

0135In the AOI interval between 0° and 45°, the hue angle h increases slightly between 0° and 30° from about h=10° to just over h=16° and then decreases more steeply to h=20° at AOI=40° (<figref idref="f0014">Figure 25</figref>).

0136For a residual reflex color red, the change Δh of the hue angle h between 0°≤AOI≤30° is preferably at most Δh=6°, particularly preferably at most Δh=5.3°. The change Δh of the hue angle h between 0°≤AOI≤45° is preferably Δh=20°, particularly preferably at most Δh=16.8°.

0137In the AOI interval between 0° and 45°, the curves of the photopic reflectance Rv and the scotopic reflectance Rv` practically do not vary between 0° and 20° and both start at 0.35, while the curve (Rv) increases more than the curve (Rv`). They increase between AOI=30° to 45° to a value of 1 (<figref idref="f0014">Figure 26</figref>).

0138The following Table 1 shows examples of the layer thicknesses of a layer system with four layer packages 42, 44, 46, 48:<tables id="tabl0001" num="0001"><table frame="all"><title>Table 1 Layer thicknesses of partial layers 60, 62 for different residual reflex colors.</title><tgroup cols="7"><colspec colnum="1" colname="col1" colwidth="10mm" /><colspec colnum="2" colname="col2" colwidth="10mm" /><colspec colnum="3" colname="col3" colwidth="30mm" /><colspec colnum="4" colname="col4" colwidth="28mm" /><colspec colnum="5" colname="col5" colwidth="29mm" /><colspec colnum="6" colname="col6" colwidth="29mm" /><colspec colnum="7" colname="col7" colwidth="27mm" /><thead valign="top"><row><entry /><entry /><entry /><entry>Blue Thickness [nm]</entry><entry>Green Thickness [nm]</entry><entry>Yellow Thickness [nm]</entry><entry>Red Thickness [nm]</entry></row></thead><tbody><row><entry /><entry /><entry>Air</entry><entry /><entry /><entry /><entry /></row><row><entry /><entry>34</entry><entry>top layer</entry><entry /><entry /><entry /><entry /></row><row><entry morerows="2">48</entry><entry>62</entry><entry>SiO₂</entry><entry>81,7</entry><entry>70,7</entry><entry>62</entry><entry>71</entry></row><row><entry>64</entry><entry>functional layer</entry><entry>12</entry><entry>16,6</entry><entry>12</entry><entry>12</entry></row><row><entry>60</entry><entry>Ta₂O₅</entry><entry>119,2</entry><entry>103</entry><entry>136</entry><entry>72</entry></row><row><entry morerows="1">46</entry><entry>62</entry><entry>SiO₂</entry><entry>17,6</entry><entry>19,1</entry><entry>9</entry><entry>13,48</entry></row><row><entry>60</entry><entry>Ta₂O₅</entry><entry>27,4</entry><entry>24,6</entry><entry>72</entry><entry>3,5</entry></row><row><entry morerows="1">44</entry><entry>62</entry><entry>SiO₂</entry><entry>49,7</entry><entry>75,9</entry><entry>6,7</entry><entry>4</entry></row><row><entry>60</entry><entry>Ta₂O₅</entry><entry>20,1</entry><entry>12</entry><entry>62</entry><entry>29,4</entry></row><row><entry morerows="1">42</entry><entry>62</entry><entry>SiO₂</entry><entry>54,9</entry><entry>64,3</entry><entry>28,8</entry><entry>41</entry></row><row><entry>60</entry><entry>Ta₂O₅</entry><entry>7,5</entry><entry>5</entry><entry>20</entry><entry>5</entry></row><row><entry /><entry>32</entry><entry>intermediate layer</entry><entry /><entry /><entry /><entry /></row><row><entry /><entry>20</entry><entry>substrate</entry><entry /><entry /><entry /><entry /></row></tbody></tgroup></table></tables>

0139It can be seen that, with the same layer materials of the high-refractive index sublayer 60 closer to the substrate and the low-refractive index sublayer 62 further away from the substrate, the color of the residual reflection can be achieved simply by changing the layer thicknesses of the sublayers 60, 62. For example, for all high-refractive index sublayers 60 Ta₂O₅ and for the low-refractive sublayers 62 SiO₂.

0140Only in the layer package 48 furthest from the substrate is a preferably low-refractive functional layer 64 arranged between the first and second partial layers 60, 62. This serves, for example, to increase the electrical conductivity and/or to equalize mechanical stress and/or as a diffusion barrier.

0141The difference between the results of layer systems 10 according to the invention with the residual reflex colours blue, green, yellow, red and the residual reflex colour of a known layer system from the prior art for a viewing angle interval of 0° to 30° is shown in<figref idref="f0015">Figure 27</figref> in a representation in polar coordinates.<figref idref="f0016">Figure 28</figref> shows the same representation with a larger viewing angle interval from 0° to 45°.

0142As can be seen from the two figures, different reflex colors with high color constancy can be achieved. The color of the residual reflection is determined by the relatively low value of the chroma C* of at most 16, particularly preferably of at most 14, at the upper limit of the viewing angle AOI.

0143In these two polar diagrams, the hue angle h is the angle of the diagram from 0° to 360° and the chroma C* is given as a radius with values ​​from 0 to 14.

0144BL denotes the course of the residual reflex colour blue, GR the course of the residual reflex colour green, GE the course of the residual reflex colour yellow, RO the course of the residual reflex colour red and PR the course of the residual reflex colour of a layer system according to the state of the art.

0145The small arrows next to the measuring points indicate the direction of the change of the viewing angle AOI from 0° to 30° or 0° to 45 ° in<figref idref="f0016">Figure 28</figref> The measured values ​​of the different residual reflex colors of the layer system 10 according to the invention practically run on straight lines through the center of the representation. The straight lines indicate the different residual reflex colors in this representation.

0146For a typical residual reflection color of green at 0° of a commercially available spectacle lens with a reflection-reducing coating according to the state of the art, the change Δh of the hue angle h between 0°≤AOI≤30° is typically Δh=127.2°. The color of the residual reflection changes from green to blue to red.

0147The change in hue angle h between 0°≤AOI≤45° is typically 161.7°.

0148For the residual reflex color blue, the change Δh in the hue angle h between 0°≤AOI≤30° is preferably at most Δh=4°, particularly preferably at most Δh=3.5°. The change Δh in the hue angle h between 0°≤AOI≤33° is preferably at most Δh=5°, particularly preferably at most Δh=4.5°.

0149For a residual reflex color green, the change Δh of the hue angle h between 0°≤AOI≤30° is preferably at most Δh=3°, particularly preferably at most Δh=2°. The change Δh of the hue angle h between 0°≤AOI≤45° is preferably at most Δh=5°, particularly preferably at most Δh=4.6°.

0150For a residual reflex color yellow, the change Δh of the hue angle h between 0°≤AOI≤30° is preferably at most Δh=1.5°, particularly preferably at most Δh=0.9°. The change Δh of the hue angle h between 0°≤AOI≤45° is preferably at most Δh=5°, particularly preferably at most Δh=4.6°.

0151For a residual reflex color red, the change Δh of the hue angle h between 0°≤AOI≤30° is preferably at most Δh=6°, particularly preferably at most Δh=5.3°. The change Δh of the hue angle h between 0°≤AOI≤45° is preferably at most Δh=20°, particularly preferably at most Δh=16.8°.

0152<figref idref="f0017">Figure 29</figref> shows a flow chart of a favorable method for designing an optical element 100 according to the invention.

0153In the method for designing an optical element 100 according to the invention, a layer design is defined in step S100. The layer design comprises at least a first material for high-refractive sublayers 60 and a second material for low-refractive sublayers 62, the number of desired layer packages 42, 44, 46, 48, or 42, 44, 46, 48, 50 with the sublayers 60, 62, the starting values ​​of the thickness of the sublayers 60, 62 and the like.

0154In step S102, target color values ​​are defined. The target color values ​​include brightness L*, chroma C* and hue angle h at least at limit values ​​for an interval of a viewing angle AOI with limit values ​​of 0° and 30°. A reflection curve can optionally be specified as an optimization specification, but it is preferred to specify the target color values ​​at the limit values ​​of the viewing angle 0° and 30°.

0155In step S104, an optimization process is carried out to vary the individual layer thicknesses until an optimization goal is reached. The optimization process then varies the individual layer thicknesses until the optimization goal (color stability) is reached.

0156Advantageously, the target colour values ​​at the limit values ​​of the interval are chosen to be the same or as similar as possible, whereby preferably only the mentioned changes Δh of the hue angle h are permitted.

0157Typically, an adapted simplex algorithm can be used as the calculation method, but other known simulation methods can also be suitable. Simulation software for such optimization methods is commercially available from various manufacturers, for example the commercial simulation software "Essential MacLeod" or other well-known simulation software for the production of optical layers.