A transmission optical system having limited ghost image visibility, a system and method for evaluating the visibility of ghost images in a transmission optical system.

Abstract

The present invention relates to a transmission optical system (1) having at least a first surface (11) and a second surface (12) with different curvatures and / or arranged to provide a refractive power. According to the invention, the first coating (21) and the second coating (22) are configured such that the optical system (1) has a colorimetric parameter for assessing the visibility of ghost images below a predetermined threshold, the colorimetric parameter for assessment being based on a total ghost image transmission coefficient of at least one ghost image formed by internal reflection between at least the first surface (11) and the second surface (12) and transmission through the optical system (1) of a light ray (30) from a light source (5), the light ray from the light source being incident on the first surface at a non-zero angle of incidence, the total ghost image transmission coefficient being integrated over the visible spectral band and depending on the spectrum of the light source.

Description

[Technical Field]

[0001] The present invention relates to a transmission optical system having low ghost image visibility, and a system and method for evaluating the visibility of ghost images in a transmission optical system.

[0002] The present invention also relates to devices and methods for reducing the visibility of ghost images in an optical system including one or more lenses.

[0003] The present invention also relates to a method for optimizing at least one anti-reflective coating of a transmitted optical system to reduce the visibility of ghost images. [Background technology]

[0004] A ghost image is generally a physical phenomenon resulting from internal reflections within one or more lenses. Ghost images are visible when viewing a light source through an optical system containing at least one lens. Ghost images are only seen with lenses that have refractive power or prisms, because these angularly separate the ghost image from the direct image of the light source. Ghost images may also appear when a first lens is clipped onto another spectacle lens, due to spurious reflections between the surface of the first lens and the surface of the other lens.

[0005] Ghost images, depending on their color and intensity, can cause discomfort to the wearer.

[0006] Numerous publications describe devices and methods for manufacturing optical systems with anti-reflective coatings that provide comfort under specific conditions, such as nighttime driving, and limit or avoid ghost images from light sources.

[0007] However, the visibility of ghost images depends on several parameters, including the reflectivity and transmittance characteristics of convex and concave surfaces, as well as the absorptivity of the substrate and the light source spectrum.

[0008] U.S. Patent No. 5,193,028 discloses a transmission optical system comprising a plurality of transmission optical elements having at least two boundary reflectors, wherein the at least two boundary reflectors form a ghost image when light is reflected by the at least two boundary reflectors. The two boundary reflectors are coated with two anti-reflective coatings having complementary reflection spectral curves to eliminate the ghost image over a wide wavelength range.

[0009] The publication “Ophthalmic lenses and dispensing”, pages 75-87, 2008-01-01, Elsevier, XP055039205 discloses multilayer and broadband anti-reflective coatings that prevent ghost images from obstructing the view through the optical system, based on a calculated or measured reflectance spectrum in the range of 400-700 nm, under conditions where light is incident and reflected at a perpendicular angle of incidence. U.S. Patent Application Publication 2020 / 0284962 relates to a method, system and composition for reducing glare that is actually present and perceived when viewed through a partially transparent material by incorporating a light absorber into the partially transparent material.

[0010] In lenses equipped with anti-reflective coatings, each anti-reflective coating may be defined using colorimetric parameters expressed in CIE color coordinates (C, h°, Rv), where C represents chroma, h represents hue angle, and Rv represents color lightness, based on the sensitivity of the human eye by a CIE 1931 photopic observer and the D65 standard reference light source.

[0011] The visibility of ghost images is related to the phenomenon of a few or more internal reflections. Furthermore, in optical systems with three or more surfaces, the number of ghost images increases with each additional surface.

[0012] Empirically, applying an anti-reflective coating to at least one surface of the optical system can reduce the visibility of ghost images. However, using conventional anti-reflective coatings on all surfaces is not sufficient to predictively avoid ghost images.

[0013] There is no method for numerically evaluating the visibility of ghost images in a transmission optical system having two or more surfaces.

[0014] There is no method to optimize one or more coatings in a transmission optical system to minimize the visibility of ghost images. [Overview of the Initiative] [Problems that the invention aims to solve]

[0015] Therefore, there is a need for criteria, systems, and / or methods for evaluating the visibility of ghost images in a transmission optical system having two or more surfaces.

[0016] There is also a need for methods to optimize anti-reflective coatings in optical systems having two or more surfaces in order to avoid or limit the visibility of ghost images. [Means for solving the problem]

[0017] Accordingly, one object of the present invention is to provide a transmission optical system having at least a first surface and a second surface, wherein the first surface and the second surface have different curvatures and / or the first surface and the second surface are arranged to provide a refractive index, the transmission optical system having a colorimetric parameter for evaluating the visibility of a ghost image that is lower than a predetermined threshold for the visibility of a ghost image, the colorimetric parameter for evaluation being based on the total ghost image transmission coefficient of at least one ghost image formed by internal reflection of a ray from a light source between at least the first surface and the second surface and transmission through the optical system, the ray from the light source being incident on the first surface at a non-zero angle of incidence, the total ghost image transmission coefficient being obtained by integrating the ghost image transmittance of the at least one ghost image over the visible spectral band, and depending on the spectrum of the light source and the spectral light efficiency of a CIE1964 photopic observer.

[0018] According to a particular advantageous aspect, a transmissive optical system having exactly two surfaces with different curvatures, wherein each ghost image is formed by internal reflection of light rays between the two surfaces of the optical system and transmission through the optical system, the transmissive optical system has a ghost image transmittance calculated using the following formula: T(λ, 15°) = T Cx (λ, 15°).R BCc (λ, α).R BCx (λ, α).T Cc (λ, α).(T int (λ, α)) 3 where λ represents the wavelength of the visible spectrum, the incident angle of the light source is set to 15 degrees, and corresponds to the refraction angle α inside the substrate supporting the first surface and the second surface, T Cx (λ, 15°) represents the spectral transmittance of the first surface at the incident angle, R BCc (λ, α) represents the spectral reflectance of the second surface regarding the reflection occurring inside the substrate medium at the refraction angle α, R BCx (λ, α) represents the spectral reflectance of the first surface regarding the reflection occurring inside the substrate medium, T Cc (λ, α) represents the spectral transmittance of the second surface, and T int (λ, α) represents the spectral transmittance between the first surface and the second surface.

[0019]

Number

Number

[0020] According to a particular aspect, the first surface includes a first coating, and the second surface includes a second coating.

[0021] In certain embodiments, the transmission optical system further includes at least another surface having another coating, and each pair of two different surfaces having pairs of different curvatures from the first surface, the second surface, and the at least another surface forms a specific ghost image, and each specific ghost image has a specific ghost image transmittance, and the first coating, the second coating, and the other coating are configured such that each specific ghost image has an overall ghost image transmittance coefficient lower than a predetermined threshold.

[0022] According to a particular embodiment, the ghost image transmittance of the entire ghost image formed by the superposition of different ghost images is given by the following formula:

number

[0023] Alternatively, according to another particular embodiment, the transmission optical system includes at least another surface including another coating, where each pair of two different surfaces having the same curvature from the first surface, the second surface and at least another surface forms the same particular ghost image component having component ghost image transmittance, and the first coating, the second coating and the other coating are configured such that, for the same particular ghost image, the total ghost image transmittance coefficient is calculated as a function of the sum of different components of the same particular ghost image transmittance and is below a predetermined threshold.

[0024] In particular, the total ghost image transmission coefficient for each ghost image is given by the following formula:

number

number

[0025] According to a particular embodiment, the total ghost image transmission coefficient is further based on the number of surfaces of the transmission optical system, the transmission coefficient of each surface, and the transmission coefficient of each substrate supporting the surfaces of the transmission optical system.

[0026] For example, the light source is a light-emitting diode with a color temperature ranging from 2700 Kelvin to 6000 Kelvin.

[0027] In a particular embodiment, a predetermined threshold for the visibility of the ghost image is less than 0.010%, preferably 0.007%.

[0028] Preferably, the transmission optical system includes at least one anti-reflective coating on the first surface and / or the second surface.

[0029] According to a particular embodiment, at least one anti-reflective coating includes a pair of coatings comprising a first anti-reflective coating on a first surface and a second anti-reflective coating on a second surface, the pair of coatings being selected from the following pairs: blue anti-reflective coating and copper anti-reflective coating, blue anti-reflective coating and green anti-reflective coating, orange mirror and blue anti-reflective coating, blue mirror and copper anti-reflective coating, green anti-reflective coating and copper anti-reflective coating, copper anti-reflective coating and copper anti-reflective coating.

[0030] According to a particular embodiment, at least one anti-reflective coating includes a pair of coatings comprising a first anti-reflective coating on a first surface and a second anti-reflective coating on a second surface, wherein the pair of coatings includes at least one of a copper anti-reflective coating, a green anti-reflective coating, and an orange mirror coating.

[0031] Advantageously, the copper anti-reflective coating comprises a multilayer stack including at least six layers, the multilayer stack including, along the direction away from the substrate, a silicon dioxide layer having a thickness of about 150 nm, a zirconium dioxide layer having a thickness of 14 nm to 16 nm, a silicon dioxide layer having a thickness of 28 nm to 32 nm, a zirconium dioxide layer having a thickness of 87 nm to 93 nm, a tin oxide or indium tin oxide layer having a thickness of about 6.5 nm, and a silicon dioxide layer having a thickness of 71.7 nm to 77 nm.

[0032] According to a particular embodiment, the pair of coatings includes a copper anti-reflective coating and at least one of a blue mirror or a blue anti-reflective coating.

[0033] A further object of the present invention is to provide a method for evaluating the visibility of a ghost image of a transmission optical system having at least a first surface and a second surface, wherein the first surface and the second surface have different curvatures and / or the first surface and the second surface are arranged to provide a refractive index.

[0034] The above objective is achieved by providing a method for evaluating the visibility of a ghost image, comprising the step of determining a colorimetric parameter for evaluating the visibility of the ghost image, wherein the colorimetric parameter for evaluation is based on the total ghost image transmission coefficient of at least one ghost image formed by internal reflection and transmission through an optical system of light rays from a light source between a first surface and a second surface, the light rays from the light source are incident on the first surface at a non-zero angle of incidence, the total ghost image transmission coefficient is obtained by integrating the ghost image transmittance of the at least one ghost image over the visible spectral band, and depends on the spectrum of the light source and the spectral light efficiency of a CIE1964 photopic observer.

[0035] The above objective is achieved, according to the present invention, by providing a method for optimizing at least one anti-reflective coating of a transmissive optical system having at least a first surface and a second surface, wherein the first surface comprises a first coating and the second surface comprises a second coating, at least one of the first coating and the second coating is an anti-reflective coating, the first surface and the second surface have different curvatures and / or the first surface and the second surface provide refractive power.

[0036] The above objective is to provide a method for optimizing at least one anti-reflective coating according to the present invention, a) A step of determining a colorimetric parameter for evaluating the visibility of a ghost image, wherein the colorimetric parameter for evaluation is based on the total ghost image transmission coefficient of at least one ghost image formed by internal reflection and transmission through an optical system of light rays from a light source between a first surface and a second surface, wherein the light rays from the light source are incident on the first surface at a non-zero angle of incidence, and the total ghost image transmission coefficient is obtained by integrating the ghost image transmittance of the at least one ghost image over the visible spectral band, and depends on the spectrum of the light source and the spectral light efficiency of a CIE1964 photopic observer. b) A step of modifying the structure and / or composition of the first coating and / or the second coating, c) Repeat steps a) and b) until the transmission optical system has a colorimetric parameter for evaluating the visibility of ghost images that is lower than a predetermined threshold for the visibility of ghost images. This is achieved by providing a method that includes [a specific component].

[0037] A further object of the present invention is to provide a system for evaluating the visibility of a ghost image of a transmission optical system having at least a first surface and a second surface, wherein the first surface and the second surface have different curvatures and / or the first surface and the second surface are arranged to provide a refractive index.

[0038] According to this disclosure, the system for evaluating the visibility of a ghost image includes a processor configured to determine a colorimetric parameter for evaluating the visibility of the ghost image, the colorimetric parameter for the evaluation being based on the total ghost image transmission coefficient of at least one ghost image formed by internal reflection and transmission through an optical system of a ray from a point light source, wherein the ray from the light source is incident on the first surface at a non-zero angle of incidence, the total ghost image transmission coefficient is obtained by integrating the ghost image transmittance of the at least one ghost image over the visible spectral band, and depends on the spectrum of the light source and the spectral light efficiency of a CIE1964 photopic observer.

[0039] According to a particular embodiment, the transmission optical system has a third surface and / or a fourth surface, and the system for evaluating the visibility of the ghost image includes a processor configured to determine the total ghost image transmission coefficient for each ghost image formed by the internal reflection of light rays between all pairs of surfaces having the same set of two different curvatures, the total ghost image transmission coefficient depends on the sum of the ghost image transmission components for all pairs of surfaces having the same set of two different curvatures.

[0040] In another embodiment, the transmission optical system has a third surface and / or a fourth surface, and the system for evaluating the visibility of the ghost images includes a processor configured to determine the ghost image transmission coefficient of each ghost image formed by the internal reflection of light rays between each pair of surfaces having two different curvatures, the total ghost image transmission coefficient depends on the sum of the ghost image transmission coefficients of each ghost image.

[0041] The following description, with reference to the accompanying drawings, should make the components of the present invention and the methods by which they can be achieved clear. The present invention is not limited to the one or more embodiments shown in the drawings. Accordingly, where reference numerals follow features mentioned in the claims, such numerals are included solely for the purpose of improving the understanding of the claims and should not be understood as limiting the scope of the claims.

[0042] Herein, the following brief description is referenced in relation to the attached drawings and detailed descriptions, and similar reference numbers represent similar parts. [Brief explanation of the drawing]

[0043] [Figure 1] This shows a cross-sectional view of the internal reflection of incident light rays at the initiation of the formation of a spectacle lens and ghost image. [Figure 2] This describes a system for observing the ghost image of a light source through an optical system. [Figure 3] Examples of direct and ghost images of the same light source as seen through an optical system are shown. [Figure 4] This diagram shows a cross-sectional view of the two-lens optical system and the multiple internal reflections of the incident light ray at the initiation of ghost image formation. [Figure 5A] This shows each ghost image formed by internal reflection between pairs of surfaces in a two-lens optical system. [Figure 5B] This shows each ghost image formed by internal reflection between pairs of surfaces in a two-lens optical system. [Figure 5C] This shows each ghost image formed by internal reflection between pairs of surfaces in a two-lens optical system. [Figure 5D]This demonstrates internal reflection between surfaces with the same radius of curvature that do not generate separate ghost images. [Figure 6] This shows an example of a ray tracing simulation of ghost image formation in a two-lens optical system. [Figure 7] The various parameters used to determine the ghost image transmittance (angle of incidence, angle of refraction, transmittance coefficient and reflection coefficient of the two-surface optical system) are shown. [Figure 8] The spectral efficiency for different types of observers under daytime and nighttime visual acuity conditions is shown, respectively. [Figure 9] For various two-surface optical systems, the evaluation results of the visibility of ghost images are shown as a logarithmic function of the total ghost image transmittance. [Figure 10] For various two-surface optical systems, correlation curves between contrast and the logarithm of total ghost image transmittance are shown. [Figure 11] This diagram schematically illustrates the formation of ghost images due to internal reflections between the four surfaces of the optical system and the first surface labeled A and the fourth surface labeled D. [Figure 12] This diagram schematically illustrates the formation of ghost images due to internal reflections between the four surfaces of the optical system and the second surface labeled B and the fourth surface labeled D. [Figure 13] This diagram schematically illustrates the formation of ghost images due to internal reflections between the four surfaces of the optical system and the third surface labeled C and the fourth surface labeled D. [Figure 14] The present disclosure shows various two-lens optical systems in which at least one of the coatings is optimized to reduce the visibility of ghost images. [Figure 15] The present disclosure shows various two-lens optical systems in which at least one of the coatings is optimized to reduce the visibility of ghost images. [Figure 16] The present disclosure shows various two-lens optical systems in which at least one of the coatings is optimized to reduce the visibility of ghost images. [Figure 17]The present disclosure shows various two-lens optical systems in which at least one of the coatings is optimized to reduce the visibility of ghost images. [Modes for carrying out the invention]

[0044] In the following description, the drawings are not necessarily to scale, and certain features may be shown in a generalized or schematic form for clarity and conciseness or for informational purposes. In addition, while the creation and use of various embodiments are discussed in detail below, it should be understood that many inventive concepts are provided that can be embodied in a variety of situations as described herein. The embodiments discussed herein are merely representative and do not limit the scope of the invention. It will also be apparent to those skilled in the art that all technical features defined in relation to a process can be replaced individually or in combination with those of a device, and conversely, all technical features defined in relation to a device can be replaced individually or in combination with those of a process.

[0045] Devices and Processes Figure 1 shows a cross-sectional view of an optical system consisting of a single spectacle lens 1 having a first surface 11 and a second surface 12. The first surface 11 and the second surface 12 have different curvatures and / or are arranged to provide a refractive power. Generally, in a spectacle lens, the first surface 11 is convex and the second surface 12 is concave. Figure 1 also shows a point light source 5 that generates an incident ray 30 directed towards the spectacle lens 1 and the two principal optical paths of the ray passing through the spectacle lens 1. When transmitted light passes directly through the first surface 11 and the second surface 11 of the lens and through the lens substrate once, a directly transmitted ray 40 is formed. When incident light undergoes internal reflection at the second surface 12 and the first surface 11 of the lens and then passes through the lens substrate twice through the second surface 12, a primary reflected ray 41 is formed. Ghost images are formed under certain conditions. In particular, in the case of a point light source 5, if the angle of incidence of the incident ray 30 on the first surface 11 is different from zero, i.e., non-zero, the directly transmitted ray 40 and the primary reflected ray 41 tend to propagate along their respective angularly separated directions, forming two different images of the light source 5. The direct image 6 is formed along the direction of the directly transmitted ray 40. The ghost image 7 of the light source 5 is formed along the direction of the primary reflected ray 41. Thus, the formation of a ghost image is a physical phenomenon resulting from internal reflection within the lens 1. Ghost images are only seen in lenses with refractive power or prisms because they angularly separate the ghost image 7 from the direct image 6 of the light source through the optical system.

[0046] Figure 2 shows a system for observing ghost images. The ghost image 7 is visible when a point light source 5 is viewed through lens 1. The point light source 5 is, for example, a light-emitting diode, i.e., an LED. The presence of ghost images can cause discomfort to the wearer, depending on the color and intensity of the ghost image.

[0047] Figure 3 shows an example of a direct image 6 and a ghost image 7 formed using an LED light source and a lens with a highly saturated blue coating on both sides. The direct image 6 is clearly and sharply visible against a dark background. The ghost image 7 is less sharp and appears colored, appearing blue in this case, for example.

[0048] Because light reflects off each surface of the lens, the colorimetric properties of the coatings applied to each surface affect the ghost image. Here, we will refer to the first coating 21 on the first surface 11 and the second coating 22 on the second surface 12, respectively. The first and second coatings are selected from the following coatings: anti-reflective coatings, mirror coatings, and hard coats. Each coating can consist of a single layer or a multilayer stack.

[0049] It is desirable to maximize the visible transmittance of the direct image while minimizing the visibility of the ghost image.

[0050] However, the relationship between the colorimetry of the first and second coatings and the visibility of ghost images is unclear. The colorimetry of a coating can be defined using three-dimensional color coordinates such as (Rv, C, h), where h represents the hue angle, C represents the saturation, and Rv represents the lightness of the color, based on the sensitivity of the human eye (Rv is derived from CIE1931, and saturation and hue are derived from CIE1964 and based on a D65 light source). When considering a combination of two anti-reflective coatings, there are many colorimetric parameters for each coating: Rv, saturation, and hue angle.

[0051] Empirically, it has been observed that the visibility of ghost images differs significantly depending on the combination of anti-reflective coatings on the convex and concave surfaces of a single lens.

[0052] According to the first approximation, for a single lens, the visibility of the ghost image depends on both the Rv of the first surface (convex) 11 and the Rv of the second surface (concave) 12. However, there exists a lens configuration in which the convex surface has the same coating and the concave surface has two different coatings with the same Rv value, and such a lens configuration results in significantly different ghost images.

[0053] Here, consider a lens having, for example, a first coating 21 consisting of a highly saturated blue coating and a second coating on the back surface. If the second coating 22 consists of a blue anti-reflective coating with an Rv of 0.96, the ghost image will appear dark blue. However, if the second coating 22 consists of a green anti-reflective coating similarly having an Rv of 0.96, the ghost image will be darker and almost invisible against a dark background.

[0054] Furthermore, in this specification, anti-reflective color is generally calculated using the D65 standard light source, which corresponds to daylighting. However, the D65 standard light source does not appear to be suitable for observing ghost images formed from point light sources against generally dark backgrounds.

[0055] When an ophthalmic system has three or more surfaces (N≧2) and one of the surfaces has a different curvature, multiple ghost images become more likely to be seen due to multiple internal reflections between these surfaces.

[0056] In an ophthalmic system with two surfaces, there exists a single ghost image, or primary ghost image, formed by double reflection within the system. From there, the number of ghost images increases with each additional surface. Therefore, a system with N≧2 surfaces can have (N-1)! primary ghost images.

[0057] Ophthalmic systems with three or more surfaces typically consist of a combination of a first lens 1 and a second lens 2, though not limited to these. Generally, the first lens is a planar lens, and the second lens is a convex lens. Such combinations may be intended to provide prescriptions for planar lenses (such as E-chromic lenses or sunglasses). Alternatively, such combinations may be intended to provide additional functions, whether temporary or temporary, on top of refractive lenses (such as a set of clips for daytime or nighttime driving, or augmented reality lenses).

[0058] Among ophthalmic systems with three or more surfaces, most systems have four surfaces composed of a specific combination of surface curvature radii. Generally, all but one surface are parallel to each other or have the same (or nearly the same) radius of curvature. Using the same radius of curvature at the interface of two clipped lenses is justified by mechanical constraints.

[0059] For example, as shown in Figure 4, the ophthalmic system consists of a first lens 1 and a second lens 2. The first lens 1 has a first convex surface 11 having a first radius of curvature denoted as R1, and a second concave surface 12 having the same radius of curvature R1. The second lens 2 has a third convex surface 13 having the same first radius of curvature R1, and a fourth concave surface 14 having a second radius of curvature denoted as R2, which is different from R1.

[0060] In the case of an ophthalmic system having three or more surfaces, several ghost images are formed. A point light source 5 generates an incident ray 30 directed onto the first lens 1 of the ophthalmic system at a non-zero angle of incidence. Figure 4 shows the principal optical path of the ray through the first lens 1 and the second lens 2. When the transmitted light directly passes through the first surface 11 and the second surface 12 of the first lens 1 and the third surface 13 and the fourth surface 14 of the second lens 2, a directly transmitted ray 40 is formed. The directly transmitted ray 40 passes once through the substrate of the first lens 1 and once through the substrate of the second lens 2. After the incident light undergoes internal reflection at the fourth surface 14 and the third surface 13 of the second lens 2, it passes once through the substrate of the first lens 1 via the fourth surface 14 and twice through the substrate of the second lens 2, forming a primary reflected ray 43 (Figure 5A). When the incident light undergoes internal reflection at the fourth surface 14 of the second lens 2 and the second surface 12 of the first lens, and then passes through the third and fourth surfaces of the second lens 2, another primary reflected ray 42 is formed (Figure 5B). Therefore, this other primary reflected ray 42 passes through the substrate of the first lens 1 once and through the substrate of the second lens 2 three times. When the incident light undergoes internal reflection at the fourth surface 14 of the second lens 2 and the first surface 11 of the first lens 1, and then passes through the second, third, and fourth surfaces, yet another primary reflected ray 41 is formed (Figure 5C). Therefore, yet another primary reflected ray 41 passes through the substrate of the first lens 1 three times and through the substrate of the second lens 2 three times. The primary reflected rays 43, 42, and 41 are parallel to each other.

[0061] This particular configuration yields several results regarding ghost images. Ghost images resulting from the same combination of two different radii of curvature R1 and R2 (Figures 5A, 5B, and 5C) all combine into a primary ghost image (Figure 6), which is more visible than when they are distinguishable.

[0062] Some ghost images occur between parallel surfaces (having the same radius of curvature R1) and are therefore merged into the direct image and become invisible. For example (see Figure 5D), reflected light rays 50 are formed by internal reflection at the second surface 12 and the first surface 11, pass through the second surface 12 of the first lens, and are transmitted through the third and fourth surfaces of the second lens 2. The reflected light rays 50 generate a ghost image. However, since the directly transmitted light rays 40 and the reflected light rays 50 propagate along parallel directions, this ghost image is superimposed on the direct image of the point light source 5.

[0063] Figure 6 shows an example of a ray tracing simulation of ghost image formation in a two-lens optical system without coatings, as shown in Figures 4 and 5. The simulation software used here is Zemax OpticStudio software. The light source used in the simulation is 40 mm square and is placed 500 mm away from the eye at an incident angle of 15 degrees from the optical axis. The light source has three equally weighted wavelengths: 486 nm, 588 nm, and 656 nm. The light source has an LED with a color temperature of approximately 4000 K. Three ghost images formed by internal reflections at a pair of surfaces with different radii of curvature (R1=193 mm and R2=117 mm) in the two-lens optical system are combined into a single ghost image 7 that is spatially offset from the primary image 6. Other stray light 8 is also generated. However, this remaining stray light 8 is virtually invisible to the human eye.

[0064] In the case of an ophthalmic system with a single lens, applying a standard anti-reflective coating to two surfaces (N=2) may be sufficient to reduce the visibility of ghost images.

[0065] However, in the case of an ophthalmic system with two lenses (as in the previous example, having three surfaces with the same radius of curvature R1 and one surface with a different radius of curvature R2), three ghost images of similar intensity are superimposed, so the sum of the transmittances of all three ghost images is approximately three times greater. However, applying a normal anti-reflective coating to all surfaces is generally not sufficient to reduce the visibility of the resulting ghost images below the visibility threshold. More generally, different combinations of radii of curvature result in different numbers of superimposed ghost images.

[0066] Therefore, the purpose of this disclosure is to propose other criteria for evaluating the visibility of ghost images.

[0067] More precisely, we propose appropriate colorimetric parameters for evaluating the visibility of ghost images. These new parameters are based on calculations of ghost image spectra and include the light source spectrum and the CIE1964 observer, which has been found to be more relevant than the conventional CIE1931 observer. Below, we specify the calculation method for ophthalmic systems with at least two surfaces.

[0068] A new parameter, the total ghost image transmission coefficient (hereinafter referred to as T). GI (Denoted as ) represents the ghost image transmittance in a two-surface optical system or the total ghost image transmittance in an optical system with three or more surfaces, supported by a strong correlation with perception, evident in results from studies by untrained observers and other studies by trained observers. For perceptual data, criteria regarding the risk of ghost images in combinations of anti-reflective coatings can be established and incorporated in the anti-reflective design process for optical systems with at least two surfaces. A numerical threshold for the visibility of ghost images is also determined.

[0069] 1. T in an optical system with two surfaces GI Definition and calculation To calculate the colorimetric parameters, we first use the transmittance spectrum of the ghost image. The transmittance spectrum allows us to obtain descriptive information about the color of the ghost image. From the transmittance spectrum, we derive numerical parameters that concisely and accurately describe and evaluate the ghost image.

[0070] Currently, ghost images cannot be measured directly, and because they require extremely specialized spectroscopic analysis equipment and acquisition conditions, they are modeled instead.

[0071] A numerical simulation tool for calculating the transmittance spectrum of ghost images was developed on MATLAB. The transmittance spectrum of ghost images can also be calculated using the vStack function of Macleod software (available in Macleod Enhanced Edition).

[0072] The numerical simulation was based on the following assumptions, schematically shown in Figure 7: - Approximate them as two parallel surfaces (without considering lens correction), - Consider the absorption and thickness of the substrate. - Calculate using an incidence angle of 15 degrees. Based on.

[0073] The angle of incidence is set to 15 degrees, representing the average observed angle. However, to suit a more specific set of conditions, any non-zero angle of incidence, preferably 5 to 30 degrees, may be used as needed.

[0074] The color of the ghost image is calculated using the transmittance of the ghost image in the spectral range of 380 to 780 nm.

[0075] The reference light source for color calculations is usually D65, the standard light source representing daylight. However, ghost images require a point light source, and since "daylight" is not a point light source, they are not observed under daylight. Sunlight is also generally unsuitable because people do not usually look directly at the sun.

[0076] Therefore, the reference light source selected herein is a specific light source used and measured for observation. For example, the reference light source is any point light source, such as an LED with a color temperature between 2700 Kelvin and 6000 Kelvin, particularly a 4000K LED, or a filament lamp, halogen lamp, or even the sun. In this specification, a point light source is a light source that appears small compared to the observer's field of view through a lens. For example, the angular size of the light source is more than 5 to 10 times smaller than the observer's entire field of view. In the calculations performed in this disclosure, the light source is approximated as a point light source, as intended by the usual definition in optics.

[0077] The colors are calculated using the following tools: Matlab (also available as a commercially available toolbox) and Macleod, which has built-in color calculation capabilities.

[0078] In the CIE XYZ color coordinate system, the R of anti-reflective coatings V This corresponds to the Y tristimulus value (value representing brightness) calculated from the reflectance of the anti-reflective coating for a second-degree field observer (CIE1931 observer). In this specification, T GI Instead, it is defined as the Y tristimulus value of the ghost image transmittance calculated with a 10-degree field observer (i.e., the CIE1964 observer). The 10-degree field observer is an updated version of the 2-degree field observer that provides corrections in the blue wavelength range and is CIE recommended for color calculations. V Similarly, T GI It is expressed as a percentage. In other words, T GI This corresponds to a colorimetric parameter that represents the luminance of the ghost image's color based on the sensitivity of the human eye derived from the CIE1964 photopic observer and the spectrum of a point source (instead of the D65 standard reference light source).

[0079] Figure 8 shows the spectral efficiency (in arbitrary units, SI) for various types of observers under daytime visual acuity (i.e., photopic) and nighttime visual acuity (i.e., scotopic) conditions, respectively. Curve 51 shows the spectral efficiency of a photopic CIE1931 observer, i.e., a 2-degree field observer, under daytime visual acuity conditions. Curve 52 shows the spectral efficiency of a scotopic CIE1951 observer, i.e., a 2-degree field observer, under nighttime visual acuity conditions. Curve 53 shows the spectral efficiency of a photopic CIE1964 observer, i.e., a 10-degree field observer, under daytime visual acuity conditions, selected in accordance with this disclosure for evaluating the visibility of ghost images.

[0080] In this case, the CIE1964 observer is chosen for two reasons. First, ghost images (in most cases) have color, and the CIE1964 observer is the recommended observer for color calculation, while the CIE1931 observer is R V These are used as standard values. The second reason relates to experimental results and will be explained later.

[0081] There is no existing standard that requires the use of the 1931 observer for the visibility criteria of ghost images. Therefore, for the reasons stated above, we propose using the 1964 observer, which appears to be more appropriate.

[0082] 2. Transmittance of ghost images or ghost image transmission coefficient (T) in a surface optical system GI (Notation) is expressed by the following equation (I):

number

number

[0083] The ghost image transmittance in a surface optical system is given by the following equation (II): T(λ,15°)=T Cx (λ,15°).R BCc (λ,α).R BCx (λ,α).T Cc (λ,α).(T int (λ,α)) 3 It is calculated using, where T Cx (λ, 15°) represents the spectral transmittance of the incident light passing through the first surface (convex surface) 11 at an incident angle of 15° corresponding to the wavelength λ, and R BCc (λ,α) represents the spectral reflectance of the light ray at the second surface (concave) 12 related to the reflection occurring inside the substrate medium at the refraction angle α, corresponding to the wavelength λ, and R BCx (λ,α) represents the spectral reflectance of the light ray at the first surface (convex surface) 11 with respect to the reflection occurring inside the substrate medium at the refraction angle α, corresponding to the wavelength λ, and T Cc (λ,α) represents the spectral transmittance of light rays passing through the second surface (concave) 12 related to transmission from the substrate medium into the air at a refraction angle α corresponding to the wavelength λ, and T int (λ,α) represents the spectral transmittance of light rays passing through the substrate supporting the first and second surfaces at a refraction angle α, corresponding to the wavelength λ (see also Figure 7).

[0084] For an angle of incidence of 15 degrees, the angle of refraction α is given by the Snell-Cartes law: n air sin(15°) = n substrate sin(α) Derived from, where n air is the refractive index of air, and n substrate This is the refractive index of the lens substrate.

[0085] For an optical system with two surfaces, the numerical threshold for the visibility of the ghost image was determined to be 0.007%. In other words, when N=2, the numerical threshold for the visibility of the ghost image is T GI It is identified as =0.007%.

[0086] The above formula T GIThe coatings on the first and second surfaces can be optimized using (15°) to obtain a ghost image transmittance that is below a numerical threshold for the visibility of 0.007% of the ghost image.

[0087] New parameter: Ghost image transmission coefficient T GI This is supported by a strong correlation with perception, evident in results from studies using untrained observers and other studies using trained observers. For perceptual data, criteria regarding the risk of ghost images in combinations of anti-reflective coatings can be established and incorporated into the anti-reflective design process for optical systems with at least two surfaces.

[0088] More precisely, the numerical threshold for the visibility of ghost images is determined by two independent methods: the first method is based on machine learning, and the second method is based on expert observation. GI It is determined empirically by a second method based on the scoring associated with it.

[0089] 2. T using machine learning GI Determining the threshold A set of 28 lenses coated with different combinations of anti-reflective coatings (i.e., AR combinations) was prepared. All other lens parameters were the same (refractive power: -2.00D, substrate: CR39, hard coat matched to a refractive index of 1.5).

[0090] On the other hand, color data is associated with each lens / AR combination. The color coordinates of the ghost image (T GI a * , b * , C * As explained in the previous paragraphs, the AR color (Rv, a) of the convex and concave surfaces is calculated for each lens using the light source spectrum S(λ) of a 4000K LED (daylight white) for observation. * , b * , C *The colors of Cx and Cc are calculated from the reflectance measurements of the exact same lens observed in the study and the spectrum of a 4000K LED. In this study, GI Similarly, Y 10 This is calculated using 1964 observers (instead of calculating RV with 1931 observers).

[0091] On the other hand, perceptual data was collected. An internal perception study was conducted using 28 lenses and 16 observers. Ghost images were observed under specific conditions. In particular, these conditions included a 4000K LED light source, a light gray background, and the use of artificial lighting in the room. This specific set of conditions was designed to be closer to everyday conditions, in contrast to other sets of conditions such as "black background and dark environment" which make ghost images more visible.

[0092] Therefore, under these conditions, there are some lenses through which the observer cannot see ghost images. The observer is asked to identify the lenses through which no ghost images were observed (i.e., in other words, no ghost images were seen).

[0093] With 16 observers and 39 lenses (some of the 28 lenses were presented twice), 16 * 39 = 624 data points were obtained, which were labeled as 0 (ghost image visible) or 1 (ghost image not visible).

[0094] Subsequently, the color data is associated with the perceptual data to build a decision tree. The color data associated with the perceptual data is combined with four different datasets, which are inputs to the decision tree algorithm. The principle of the algorithm is to be able to predict a 0 / 1 perception (whether a ghost image is visible or not) from the input data (color data) alone. Of the 624 perceptual data points, 70% (randomly selected) are used to train the algorithm and 30% are used for validation.

[0095] The dataset is as follows:

[0096] [Table 1]

[0097] The dataset that gives the most accurate and simple results is the color coordinates (T) of all ghost images. GI a * , b * , C * This is a color dataset of ghost images over ,h). The algorithm was run 11 times. All attempts were successful. GI This was considered the most relevant parameter among all the coordinates. The breakdown of these 11 tests is as follows: - 5 executions of a single condition tree, T GI >0.0071%, - 4 executions of a single condition tree, T GI >0.0063%, - Two executions of a 3-condition tree, the initial condition is T GI The percentage is >0.0071%.

[0098] The predetermined threshold was determined from the rounded weighted average of the above results (0.0068%).

[0099] This predetermined threshold is defined to be less than 0.010%, preferably less than 0.009%, more preferably less than 0.008%, and even more preferably less than 0.007%.

[0100] Each of the 11 trees correctly predicted 80-90% of all 0s and 1s in the validation data.

[0101] CIE 1931 vs. CIE 1964 Another reason for selecting the CIE1964 observer is that this opacity data shows that the CIE1964 observer provides a better correlation to observations than the CIE1931 observer.

[0102] Figure 9 shows the logarithm of the ghost image transmission coefficient log(T GIThis represents the proportion of people who did not see the ghost image in the internal study as a function of ) (the sum of "1" = "did not see" obtained relative to the total number of observers).

[0103] However, a similar graph using 1931 observers presents a large outlier corresponding to a sample containing a combination of a blue-violet cut coating on a convex surface and a high-efficiency anti-reflective coating on a concave surface. This sample is interesting and therefore important to consider.

[0104] 3.T GI and the correlation with scoring by skilled observers The external observer study replicated one of the conditions used in the internal study: an LED light source and observations against a black background in a dark environment (see also Figure 2). Twelve expert observers evaluated each of the 28 lenses on an absolute scale according to three descriptors that characterize the ghost image: contrast, sharpness, and luminosity.

[0105] Three descriptors are generated by 12 observers over two one-hour sessions. The observers are trained on the descriptors over the two one-hour sessions to provide reproducible, accurately scaled scores that can later be associated with colorimetric parameters. All 28 lenses are evaluated twice by each expert.

[0106] In summary, each lens obtains 24 individual scores for each descriptor.

[0107] The scores of each descriptor are correlated to a degree that suggests they describe the same amount. The contrast descriptor appears to account for 99% of the variation. This means that each of the other two descriptors (sharpness and luminosity) differs from the contrast descriptor by less than 1%, and using either of the other two descriptors is 99% equal. Therefore, in Figure 10, only the contrast score is used.

[0108] The data points are presented in logarithmic form. Thus, it is clear that human perception of brightness changes logarithmically with respect to physical quantities. The results are shown in contrast scores and T values, as assessed by experts. GI This clearly shows a strong correlation with the logarithm of [the function / object].

[0109] Figure 10 shows contrast score vs. T. GI This shows the logarithm of [the given value].

[0110] In Figure 10, log(T GI A linear correlation of contrast scores is observed for log(T). In particular, log(T) GI The correlation with ) is a correlation coefficient R greater than 0.85. 2 It has. In sensory analysis, R greater than 0.75 2 The coefficients are considered good. The dotted line here corresponds to the 95% prediction interval. New data points are within this dotted line with 95% probability.

[0111] log(R vCx *R vCc The correlation with ) also appears to be good, but T GI The correlation coefficient is lower, at 0.75. (Product R) vCx *R vCc Note that the parameters do not take into account the potential compensation for substrate absorption or Cx / Cc reflectance. In this case, the data were obtained only from Orma lenses with very little or no absorption, but for other substrates R vCx *R vCc The accuracy will decrease.

[0112] Two independent methods (a machine learning-based method and a skilled observer method, T GI As a conclusion of the scoring method associated with the threshold 0.007%, the ghost image transmittance associated with the threshold 0.007%, i.e., T GI The quantity provides a new criterion for the visibility of ghost images adapted to human perception. This criterion takes into account the spectrum of the light source. Observations were made with clear lenses only, but T GIThis also takes into account the absorption of the substrate. The threshold is obtained by observations under specific conditions selected to closely resemble actual indoor conditions.

[0113] Therefore, this criterion is used in the coating design process to assess the risk of ghost images for any combination of coatings.

[0114] T for an optical system having two surfaces GI The criteria are extended to optical systems with three or more surfaces, as detailed in the following sections.

[0115] 4. Calculation for systems with a surface N≧2 For this purpose, the colorimetric parameters for evaluating the visibility of ghost images are adapted to multiple ghost images. These colorimetric parameters for evaluating the visibility of ghost images enable the optimization of coating combinations on different surfaces.

[0116] First, the combinations of curvatures of the optical system are identified. Each possible combination of the two curvatures is examined based on the following rule. • Different curvature combinations (R1, R2) or (R2, R3) are considered separately. • All similar combinations of two curvatures (R1, R2) or (R2, R1) are considered together (except when another R3 radius of curvature surface exists between one of the surfaces of the combination but not between the other surfaces; in that case, that particular combination is excluded). • All combinations of the same curvature (R1, R1) are ignored, except when there is a prism between these two surfaces (in which case the two surfaces are not parallel).

[0117] This calculation model needs to be adapted to systems with three or more surfaces, taking into account the rules for the combinations of curvature described above.

[0118] In a system with three or more surfaces, if there are only two identical combinations of curvatures R1 and R2, then the T of each combination is GI These should not be considered separately, but rather, for an accurate assessment of the visibility of the ghost image, all of these specific combinations of T should be taken into consideration. GI This is the sum of the two values. This calculation excludes ghost images resulting from combinations of surfaces having the same curvature R1 and R1.

[0119] This section describes a method for calculating the total ghost image transmission coefficient for an optical system with four surfaces using two lenses. However, those skilled in the art should be able to apply this method to other optical systems with three or more surfaces.

[0120] We propose two alternative methods for calculating the total ghost image transmission coefficient for ghost images in transmission, which are formed by light reflected from exactly two surfaces of a transmission optical system.

[0121] For illustrative purposes, consider an optical system including a first lens 1 and a second lens 2, as shown in Figure 4. The first convex surface 11 has a first radius of curvature R1, the second concave surface 12 has the same radius of curvature R1, the third convex surface 13 has the same first radius of curvature R1, and the fourth concave surface 14 has a second radius of curvature denoted as R2, which is different from R1.

[0122] Method 1 Each pair of two different surfaces, among the first surface, the second surface, and at least one other surface, having pairs with different curvatures, forms a ghost image, and each ghost image has a ghost image transmittance depending on the wavelength and angle of incidence.

[0123] As schematically shown in Figures 11 to 13, in the example of a two-lens optical system, the first surface is denoted as A, the second surface as B, the third surface as C, and the fourth surface as D.

[0124] The transmission coefficient of each ghost image is calculated separately as a function of the transmittance of each ghost image.

[0125] According to the first method, the total ghost image transmittance T of the obtained ghost image GI,T is given by the following equation: T GI,T = T GI,AD + T GI,BD + T GI,CD (III) where

Number

Number

[0126] Here, since surfaces A, B, and C have the same radius of curvature R1, the corresponding ghost image transmittances: T GI,AB , T GI,AC and T GI,BC are not considered.

[0127] Here, for each surface set X of AD, BD, or CD, the calculation of each ghost image transmittance T G,X (λ, 15°) will be described.

[0128] ·T G,AD Consider the combination of the first surface A with radius of curvature R1 and the fourth surface D with radius of curvature R2. The primary ghost image is obtained from the internal reflection of the incident ray 30 on surfaces A and D that forms the primary reflected ray 41 (see Fig. 11).

[0129] The ghost image transmittance T G,ADAccording to the transmission coefficients and reflection coefficients of different interfaces and substrates through which the incident light ray 30 is sequentially reflected, refracted, or transmitted, the following formula (V): T G,AD (λ,15°) =T A (λ,15°)*T intAB (λ,α)*T B (λ,15°)*T C (λ,15°)*T intCD (λ,β)*BR D (λ,β)*T intCD (λ,β)*T C (λ,15°)*T B (λ,15°)*T intAB (λ,α)*BR A (λ,α)*T intAB (λ,α)*T B (λ,15°)*T C (λ,15°)*T intCD (λ,β)*T D (λ,15°) is calculated by, where the incident angle is set to 15 degrees, T Y (λ,15°) represents the transmission coefficient through the surface Y (Y is selected from surfaces A, B, C, and D) at an incident angle of 15 degrees according to the wavelength λ, T intAB (λ,α) represents the transmission coefficient between the surface A and the surface B at the refraction angle α within the first lens according to the wavelength λ, T intCD (λ,β) represents the transmission coefficient between the surface C and the surface D at the refraction angle β within the second lens according to the wavelength λ, BR D (λ,β) represents the rear reflection coefficient at the surface D at the refraction angle β (i.e., the reflection occurring within the substrate of the second lens) according to the wavelength λ, and BR A (λ,α) represents the rear reflection coefficient at the surface A at the refraction angle α (i.e., the reflection occurring within the substrate of the first lens) according to the wavelength λ.

[0130] The refraction angle α and the refraction angle β are based on Snell - Descartes' law: n air sin(15°)=nAB sin(α) n air sin(15°) = n CD sin(β) This is calculated by, where n AB This represents the refractive index of the substrate supporting surface A and surface B, and n CD This represents the refractive index of the substrate supporting surfaces C and D.

[0131] The model assumes all surfaces are flat and parallel, so the angle of incidence in air is always 15 degrees. In addition, the principle of regressive light assumes that the path of a light ray is the same regardless of the direction. Therefore, regardless of the actual direction, all transmittances are correctly evaluated when considering an incidence angle of 15 degrees from the air medium toward the substrate.

[0132] Therefore, the ghost image transmittance T G,AD It is calculated according to the following simplified formula. T G,AD (λ, 15°) =T A (λ,15°)*BR D (λ,β)*BR A (λ,α)*T D (λ,15°)*(T B (λ, 15°)) 3 *(T intAB (λ,α)) 3 *(T C (λ, 15°)) 3 *(T intCD (λ,β)) 3

[0133] If there is no void between surface B and surface C, but another medium exists, the angle of refraction γ inside this medium is calculated according to the Snell-Cartes law. If this medium is absorptive, the above equation is given for T at wavelength λ and angle of refraction γ. intBC It will be updated to take this into consideration.

[0134] These formulas do not take into account the polarization of light, and therefore are not exactly the same as ray tracing calculations. However, at low angles of incidence (i.e., less than 45 degrees), the effect of polarization is generally small.

[0135] Next, the ghost image transmittance T of the primary ghost image formed by internal reflection between surface A and surface D. GI、AD This is the ghost image transmittance T of the two-surface optical system. GI Similarly, ghost image transmittance T G、AD Calculated from, ghost image transmittance T GI,AD This is a single LED as a light source and a CIE1964 observer, with a ghost image transmittance of T in the CIE XYZ color system. G,AD This corresponds to the Y tristimulus values ​​(please refer to the formula above for this part).

[0136] ·T G,BD Consider the combination of a second surface B with radius of curvature R1 and a fourth surface D with radius of curvature R2. The primary ghost image is obtained from the internal reflection of the incident light ray 30 onto surfaces B and D, which forms the primary reflected light ray 42 (see Figure 12).

[0137] Ghost image transmittance T G,BD Using the same notation as above, the following equation (VI) is used, according to the transmittance coefficient and reflectance coefficient of different interfaces and substrates through which the incident light ray 30 is sequentially reflected, refracted, or transmitted: T G,BD (λ, 15°) =T A (λ,15°)*T intAB (λ,α)*T B (λ,15°)*T C (λ,15°)*T intCD (λ,β)*BR D (λ,β)*T intCD (λ,β)*T C (λ,15°)*R B (λ,15)*T C (λ,15°)*T intCD (λ,β)*T D (λ, 15°) It is calculated by, or in a simplified version, T G,BD (λ, 15°) =T A (λ,15°)*T intAB (λ,α)*T B (λ,15°)*BR D (λ,β)*R B (λ,15)*T D (λ,15°)*(TC(λ,15°)) 3 *(T intCD (λ,β)) 3 These equations are derived from the Snell-Cartes law, as described above, where the angles of refraction α and β are obtained.

[0138] Similarly, next, the ghost image transmittance T of the primary ghost image formed by internal reflection between surface B and surface D. GI、BD This is the ghost image transmittance T of the two-surface optical system. GI Similarly, ghost image transmittance T G、BD Calculated from, ghost image transmittance T GI,BD This is a single LED as a light source and a CIE1964 observer, with a ghost image transmittance of T in the CIE XYZ color system. G,BD This corresponds to the Y tristimulus values ​​(see the formula above).

[0139] ·T G,CD Similarly, consider the combination of a third surface C with radius of curvature R1 and a fourth surface D with radius of curvature R2. The primary ghost image is obtained from the internal reflection of the incident light ray 30 onto surfaces C and D, which forms the primary reflected light ray 43 (see Figure 13).

[0140] Ghost image transmittance T G,CD Using the same notation as above, the following equation (VII) is used, according to the transmittance and reflectance coefficients of different interfaces and substrates through which the incident light ray 30 is sequentially reflected, refracted, or transmitted: T G,CD (λ, 15°) =T A (λ,15°)*T intAB (λ,α)*TB (λ,15°)*T C (λ,15°)*T intCD (λ,β)*BR D (λ,β)*T intCD (λ,β)*BR C (λ,β)*T intCD (λ,β)*T D (λ, 15°) It is calculated by, or in a simplified version, T G,CD (λ, 15°) =T A (λ,15°)*T intAB (λ,α)*T B (λ,15°)*T C (λ,15°)*BR D (λ,β)*BR C (λ,β)*T D (λ,15°)*(T intCD (λ,β)) 3 That is the case.

[0141] Similarly, next, the ghost image transmittance T of the primary ghost image formed by internal reflection between surface C and surface D. GI、CD This is the ghost image transmittance T of the two-surface optical system. GI Similarly, ghost image transmittance T G、CD Calculated from, ghost image transmittance T GI,CD This is a single LED as a light source and a CIE1964 observer, with a ghost image transmittance of T in the CIE XYZ color system. G,CD This corresponds to the Y tristimulus values ​​(see the formula above).

[0142] Therefore, the total ghost image transmittance T of the ghost image formed by the superposition of the ghost images formed by the light rays 41, 42, and 43 is GI,T The ghost image transmittance T is calculated according to the above equations (V), (VI), and (VII). GI,AD , T GI,BD and T GI,CD It is calculated by the sum of (see formula (III)).

[0143] Method 2 Similar to the case of the first method, for each pair of two different surfaces having different curvatures among the first surface, the second surface, and at least one other surface, a ghost image is formed, and each ghost image has a ghost image transmittance at an incident angle of 15 degrees according to the wavelength.

[0144] Consider a two-lens optical system, where the first surface is denoted as A, the second surface is denoted as B, the third surface is denoted as C, and the fourth surface is denoted as D (see FIGS. 11 to 13).

[0145] According to the second method, the total ghost image transmittance coefficient T formed by superimposing the ghost images generated from light rays 41, 42, and 43 GI,T is given by the following equation (VIII) according to the sum of each ghost image transmittance:

Equation

Equation

[0146] As another example, consider an optical system having three surfaces. The optical system consists of, for example, highly absorbent solar glass and a refractive lens, where the solar glass is highly absorbent and cannot be given a mirror coating. The solar glass is, for example, clipped to the refractive lens. The first surface on the solar glass is denoted as A, the second surface on the solar glass is denoted as B, the third surface on the lens is denoted as C, and the fourth surface on the lens is denoted as D.

[0147] Because solar glass is highly absorbent, components related to surface A can be ignored. For example, the transmittance coefficient through solar clip is Tint = 10% or 0.1. This transmittance term is Tint 2 Furthermore, it is not significant, therefore it does not contribute significantly. Unless surface A is covered with a mirror coating, Tint 2 It has a sufficiently high reflectivity to compensate for the deficiency.

[0148] Assume that surfaces A, B, and C have the same radius of curvature, and surface D has a different radius of curvature. In this case, T G,AD This can be ignored, and the following formula: T G =T G,BD +T G,CD There is only one ghost image with a transmittance equal to T, where T G,BD and T G,CD This is calculated using equations (VI) and (VII), detailed in the previous section, either in their full or simplified versions.

[0149] The total ghost image transmission coefficient T is obtained by superimposing the ghost images generated from light rays from surfaces B, C, and D. GI,T The sum of the ghost image transmittances is given by the following equation (IX):

number

number

[0150] The same method is adapted to optical systems with three surfaces, such as a compound lens where two lenses are mounted at a common interface. This optical system has a first surface denoted as A, a second surface denoted as B, and a third surface denoted as C at the interface between the two lenses. The three surfaces A, B, and C have different radii of curvature when taken in pairs. Therefore, there are three sets of radii of curvature. This optical system with three surfaces generates three distinct total ghost images, denoted as AB, AC, and BC, respectively. The transmittance of each ghost image is calculated using the following formulas. T G,AB (λ)=T A (λ)*BR B (λ)*BR A (λ)*T B (λ)*T C (λ)*(T intAB (λ)) 3 *T intBC (λ) T G,AC (λ)=T A (λ)*BR C (λ)*BR A (λ)*(T B (λ)) 3 *T C (λ)*(T intAB (λ)) 3 *(T intBC (λ)) 3 T G,BC (λ)=T A (λ)*T B (λ)*BR C (λ)*BR B (λ)*T C (λ)*(T intBC (λ)) 3 *T intBC (λ)

[0151] The overall ghost image transmittance is calculated for each of the three ghost images using the following equations respectively.

Equation

[0152] T GI,1 , T GI,2 and T GI,3 each must be lower than a predetermined threshold regarding the visibility of the ghost image.

[0153] In summary, for an optical system having three or more surfaces, each overall ghost image is the sum of ghost image components generated by the same set of radii of curvature. An optical system having three or more surfaces can generate two or more overall ghost images. The number of overall ghost images is the same as the number of different sets of radii of curvature. Each overall ghost image must have T GI lower than a predetermined threshold regarding the visibility of the ghost image (T GI <7.10 -3 ).

[0154] The transmittance of the overall ghost image is calculated using the following equation.

Equation

[0155] Here, XY represents a pair of surfaces X and Y having the same combination of two radii of curvature, Z represents the substrate or glass plate between surface X and surface Y, and W represents each other surface different from A and B located between surface A and surface B, R represents the reflectance of the interface considered at a specific incident angle of the related array, T represents the surface transmittance at a specific incident angle of the related array, and T int represents the internal substrate transmittance at a specific incident angle of the related array. In the above equation, the surface transmittance and the internal transmittance are squared because they correspond to the round-trip of light.

[0156] T total system(λ) represents the direct transmittance corresponding to the wavelength of light rays passing through all substrates and all surfaces.

[0157] Total ghost image threshold for optical systems with three or more surfaces The threshold for the total ghost image transmission coefficient T is the same as for an optical system having only two surfaces: 0.010%, preferably 0.009%, more preferably 0.008%, and even more preferably 0.007%. GI,T Applies to this.

[0158] Third method If the incident light has a non-zero angle of incidence, even if a ghost image appears in a system of multiple central surfaces (surfaces with coincident optical axes), an additional method is the same as the above T when the angle of incidence is 0°. G (λ), T totalsystem (λ) and T GI This includes calculating the T values ​​corresponding to the ghost image, which are calculated at an incident angle of 0° and therefore should not physically exist, but are comparable to each other in order to determine the ranking of the probability of ghost image occurrence among several systems of three or more surfaces. GI Values ​​are obtained. These values ​​are not complex to calculate and can also constitute approximations of values ​​obtained at non-zero angles of incidence.

[0159] 5. Application to coating optimization Therefore, a system and method for evaluating the visibility of ghost images are provided, which can determine the visibility of ghost images in optical systems having two or more surfaces, and can propose new coating combinations that prevent ghost images from bothering the user.

[0160] Total ghost image transmission coefficient T GI,T This is optimized to be lower than a predetermined threshold, for example, less than 7‰. In particular, the total ghost image transmission coefficient is calculated using a database of known coatings.

[0161] A system and method for evaluating the visibility of ghost images also enables the creation of anti-reflective coatings that allow for the construction of optical systems having surfaces coated with this anti-reflective coating, and the optical system has a total ghost image transmission coefficient T lower than a predetermined threshold. GI、T It holds.

[0162] This system is T GI,T It is also possible to optimize the combination of anti-reflective coatings based on minimizing the total ghost image transmission coefficient so that it is lower than a predetermined threshold.

[0163] Consider an optical system consisting of two lenses, each having three surfaces 11, 12, and 13 with the same radius of curvature R1 (as shown in Figure 4), and a fourth surface 14 with a different radius of curvature R2. In this case, all possible combinations of the two surfaces are either (a)(R1, R2) or (b)(R1, R1). As explained by the combination rules above, only combination (a) produces a ghost image, while combination (b) can be ignored.

[0164] Each of the four surfaces includes a coating. However, the coating on the fourth surface 14, which has a different radius of curvature, is a first-order parameter for all individual ghost images. Therefore, in this configuration, the improvement is generally uniform whether all coatings on all three parallel surfaces 11, 12, and 13 are optimized, or only the coatings on the surface 14 with a different curvature are optimized.

[0165] In the first example (see Figure 14), the coatings on surfaces 11, 12, and 13 are fixed, and only the coating on the fourth surface 14 is modified during the optimization process according to the overall ghost image transmittance criterion of the optical system. Since the planar portion of the system (first lens 1) is provided as is (and therefore the coatings cannot be removed or changed), the coatings on surfaces 11, 12, and 13 can be fixed, or the coating on the first lens 1 is a more restrictive option, making the coating on the surface 14 of the second lens a preferred modification. This situation can occur with clear electrochromic lenses fitted with refractive add-ons (any prescription) or when clear clips are used with prescription lenses (e.g., for night driving conditions).

[0166] In the second example (see Figure 15), the coating on the fourth surface 14 is fixed, while the coatings on surfaces 11, 12, and 13 are modified during the optimization process according to the overall ghost image transmittance criterion of the optical system. In this particular configuration, all three coatings on surfaces 11, 12, and 13 contribute to the same extent as the coating on the fourth surface 14 in the first example. Therefore, modifying only one or two coatings (11, 12, and / or 13) here is less "efficient" than modifying the coating on the fourth surface, as in the first example. This situation can correspond to the same optical system as in the first example, but with different constraints on the coatings. For example, the coating on the fourth surface 14 may be fixed due to stringent E-SPF (Eye Protection Factor) requirements or crazing issues.

[0167] In the third example (see Figure 16), the entire system is designed simultaneously. All four surface coatings can be optimized simultaneously according to the overall ghost image transmittance criterion of the optical system. This configuration allows for further customization, even when there are strong constraints on the optical targets of some coatings, but at least one degree of freedom still exists in the design of each coating.

[0168] In the fourth example (see Figure 17), the first lens 1 has low transmittance (for example, a tinted lens or solar clip is added). In this case, the ghost image reflected onto the first surface 11 can be ignored due to the low transmittance of the solar lens. As a result, the internal reflection at the first surface passes through the tinted lens three times, compared to the other ghost images and direct images which pass through only once, so the corresponding ghost image transmittance is particularly reduced. This situation can correspond to the same optical system as in the previous example.

[0169] Examples of applications for the selection of anti-reflective (AR) coatings for E-chromic cells and Rx add-ons with voids. The optical system includes a first lens 1 and a second lens 2 (as shown in Figure 4). The first lens 1 includes an electrochromic cell, i.e., an E-chromic cell, which allows light to be controlled by power. The first surface 11 and the second surface 12 of the electrochromic cell have the same radius of curvature R1. The second lens 2 includes a refractive add-on or Rx add-on, the surface 13 having a radius of curvature R1, and the fourth surface having a radius of curvature R2 different from R1. A gap is located between the back surface 12 of the first lens and the surface 13 of the second lens.

[0170] The coatings on the first surface 11 and the second surface 12 of the electrochromic cell are fixed because they are supplied by the supplier. Furthermore, since the components and materials from which the electrochromic cell is made are largely unknown, the cell needs to be characterized to suit the performance of the ghost image.

[0171] The optical system has additional constraints. The UV reflectance should be low on the fourth concave surface of the add-on in order to maintain good E-SPF. The coatings on surfaces 13 and 14 should be four layers to limit crazing problems while presenting good AR optical performance (Rv < 1%) by keeping the total film thickness thin.

[0172] The coating on the third surface 13 includes a multilayer stack as shown in Table I below. In Table I, the layer numbers are labeled 1 to 5 along the direction from air to substrate, and the thickness of each layer is in nanometers.

[0173] [Table 2]

[0174] The initial design for the coating on the fourth surface 14 includes a multilayer stack as shown in Table II below.

[0175] [Table 3]

[0176] The optimized design for the coating on the fourth surface 14 includes a multilayer stack as shown in Table III below.

[0177] [Table 4]

[0178] The simulation results yielded the following performance:

[0179] [Table 5]

[0180] In Table IV, T GI1 Here, T GI,AD +T GI,BD Corresponding to T GI2 is, T GI,CD It corresponds to.

[0181] As a result of the initial coating design, a high total T of 0.0178 was achieved. GI The result was obtained. According to the simulation, the optimized coating on the concave surface 14 maintained good E-SPF performance (approximately 25) while achieving a total T GIThis can be reduced to less than the specified threshold of 0.007.

[0182] 6. System T GI Measurement First, consider an optical system that includes only one lens having a convex surface (Cx) and a concave surface (Cc).

[0183] Internal reflection ghost images cannot be measured directly and require extremely specialized spectroscopic instruments and acquisition conditions; therefore, the method described here involves independently measuring each of the elements necessary for the calculation (as explained in relation to Figure 7). T GI,T (λ,15°)=T Cx (λ,15°).R BCx (λ,α).R BCc (λ,α).T Cx (λ,α).T int (λ,α) 3

[0184] The approximate formula and measurement method are described below.

[0185] [Table 6]

[0186] SMR stands for Reflectance Measurement System, and is generally based on a spectrometer adapted to measure the reflectance of a lens at a determined angle of incidence. The Cary50 is another instrument for measuring transmittance at a perpendicular angle of incidence.

[0187] For non-absorbent coatings, we assume that the sum of the reflectance coefficient (R) and the transmittance coefficient (T) is equal to 1 (R + T = 1).

[0188] Lens back reflectivity (R BCx or R BCc Since there is no means to measure the rear reflectance (i.e., the inside of the lens) of a lens, we use the approximation that it is approximately equal to the front reflectance (i.e., the outside of the lens) at the same surface of the lens.

[0189] In the case of an absorbent substrate, the sum of the reflectance coefficient (R), transmittance coefficient (T), and absorption coefficient (A) is equal to 1 (R + T + A = 1). Therefore, T int =1-A=R tot +T tot Here, T tot This is the transmittance on both sides of the lens, and R tot This is the reflectivity of both sides of the lens. tot Although it is possible to measure the reflectance, it is not possible to measure the reflectance at 0° incidence. Therefore, the reflectance on both sides is approximated by the minimum measurement incidence angle of 10 degrees using a multi-incident SMR.

[0190] Next, the ghost image transmittance as a function of wavelength at an incident angle of 15° is calculated using the following formula as the product of the spectral quantities measured in the previous table. T G (λ,15°)=T Cx *R Cx *T Cc *RCc*T int 3

[0191] Using the measurement method described, T was applied to three commercially available lenses that have a ghost image visibility problem. GI We estimated the values ​​and summarized the results in the table below.

[0192] [Table 7]

[0193] These examples are T GI The calculation result shows that it is the most accurate quantitative representation of perception (compared to scores by expert judges) because it is based on complete and accurate information about the optical system, light source, and observer.

[0194] Next, T GI This spectrum is calculated as the Y tristimulus value for an LED light source and 1964 observers.

[0195] The following table shows the ghost image transmission coefficient T. GI We present simulation results and descriptions of each stack for several single-lens ophthalmic systems that validate the conditions for colorimetric parameters <0.007%.

[0196] [Table 8]

[0197] The structure and composition of the multilayer coating are described in detail in the table below.

[0198] [Table 9]

[0199] Other examples relating to copper anti-reflective coatings (or copper AR) correspond to Examples 9, 10, and 11 disclosed in International Publication No. 2012 / 076714 and are incorporated herein. More generally, a copper anti-reflective coating comprises a multilayer stack comprising at least six layers, the multilayer stack comprising, along the direction away from the substrate, a silicon dioxide layer having a thickness of about 150 nm, a zirconium dioxide layer having a thickness of 14 nm to 16 nm, a silicon dioxide layer having a thickness of 28 nm to 32 nm, a zirconium dioxide layer having a thickness of 87 nm to 93 nm, a tin oxide or indium tin oxide layer having a thickness of about 6.5 nm, and a silicon dioxide layer having a thickness of 71.7 nm to 77 nm.

[0200] In a particular favorable example, the optical system includes a copper anti-reflective coating on a concave surface and at least one of a blue mirror or a blue anti-reflective coating on a convex surface.

[0201] Finally, consider an optical system with three or more surfaces (N>2). Depending on the system, T GIMeasurement methods are sometimes possible. For this purpose, each term in the equation needs to be measured (see equations (III) to (VIII)). Therefore, in order to measure the reflectance and transmittance of each surface or group of surfaces, the optical system needs to be disassembled (when the optical system is made of two combined lenses). The main issue, as with two-surface optical systems, is the internal transmittance of the substrate material. In the case of a disassembled optical system, the internal transmittance can be approximated by the total transmittance and total reflectance of each lens.

[0202] While typical processes and optical systems have been described in detail herein, those skilled in the art will recognize that various substitutions and modifications are possible without departing from the scope described and defined in the appended claims.

[0203] 8. Application to optical systems having three or four surfaces This method and system are applicable to optical systems having three or more surfaces, for example, four surfaces.

[0204] In the first example, the optical system includes an electrochromic cell attached to a lens. The electrochromic cell has two surfaces having the same radius of curvature (S1, S2). The lens has a first surface with the same radius of curvature S1 and a second surface with a different radius of curvature S2. This optical system generates a single ghost image with a total ghost image transmittance formed by three components S1 and S2.

[0205] In the second example, the optical system includes a clip attached to a lens. The lens has a first surface having a first radius of curvature S1 and a second surface having a different radius of curvature S2. The clip, i.e., a planar lens, has two surfaces having the same radius of curvature (S3, S3), but different from the closest surface of the lens. As follows, this optical system generates three distinct ghost images, each ghost image having a total ghost image transmittance formed by two components: a first total ghost image generated by the two surfaces of the lens, a second total ghost image generated by the first surface of the lens and the two surfaces of the clip having the same radius of curvature (S1S3+S1S3), and a third total ghost image generated by the second surface of the lens and the two surfaces of the clip having the same radius of curvature (S2S3+S2S3).

[0206] In the third example, the optical system includes a clip attached to a flat glass. The flat glass has two surfaces having the same radius of curvature (S1, S1). The clip, which is also flat, has two surfaces having the same radius of curvature (S3, S3), but different from the surface of the lens. This optical system generates a single ghost image with a total ghost image transmittance formed by four components, as follows: a first component derived from the first surface S1 of the lens and the first surface S3 of the clip; a second component derived from the second surface S1 of the lens and the first surface S3 of the clip; a third component derived from the first surface S1 of the lens and the second surface S3 of the clip; and a fourth component derived from the second surface S1 of the lens and the second surface S3 of the clip. Finally, some aspects of the present invention are described below. [Aspect 1] A transmission optical system having at least a first surface and a second surface, wherein the first surface and the second surface have different curvatures and / or the first surface and the second surface are arranged to provide a refractive index, the transmission optical system having a colorimetric parameter for evaluating the visibility of a ghost image that is lower than a predetermined threshold for the visibility of a ghost image, the colorimetric parameter for evaluation being based on the total ghost image transmission coefficient of at least one ghost image formed by internal reflection of a ray from a light source between at least the first surface and the second surface and transmission through the optical system, the ray from the light source being incident on the first surface at a non-zero angle of incidence, the total ghost image transmission coefficient being obtained by integrating the ghost image transmittance of the at least one ghost image over the visible spectral band, and depending on the spectrum of the light source and the spectral light efficiency of a CIE1964 photopic observer. [Aspect 2] Having exactly two surfaces, the ghost image is given by the following formula: T(λ,15°)=T Cx (λ,15°).R BCc (λ,α).R BCx (λ,α).T Cc (λ,α).(T int (λ,α))3 The ghost image transmittance is calculated using, where λ represents the wavelength of the visible spectrum, the incident angle of the light source is set to 15 degrees, and corresponds to the refraction angle α inside the substrate supporting the first surface and the second surface, T Cx (λ, 15°) represents the spectral transmittance of the first surface at the angle of incidence, and R BCc (λ,α) represents the spectral reflectance of the second surface with respect to the reflection generated inside the substrate medium at the refraction angle α, and R BCx (λ,α) represents the spectral reflectance of the first surface with respect to reflections occurring inside the substrate medium, T Cc (λ,α) represents the spectral transmittance of the second surface, and T int The transmission optical system according to embodiment 1, wherein (λ,α) represents the spectral transmittance between the first surface and the second surface. [Aspect 3] The total ghost image transmission coefficient is given by the following formula:

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Claims

1. A transmission optical system having at least a first surface and a second surface, The first surface and the second surface have different curvatures, and / or the first surface and the second surface are arranged to provide a refractive index. The aforementioned transmission optical system has a colorimetric parameter for evaluating the visibility of ghost images that is lower than a predetermined threshold for the visibility of ghost images, and the predetermined threshold for the visibility of ghost images is less than 0.007%. The colorimetric parameters for the evaluation are based on the total ghost image transmission coefficient of at least one ghost image formed by internal reflection of light rays from the light source between at least the first surface and the second surface and transmission through the transmission optical system, wherein the light source is a light-emitting diode having a color temperature in the range of 2700 Kelvin to 6000 Kelvin. The light ray from the light source is incident on the first surface at a non-zero angle of incidence. The total ghost image transmission coefficient is obtained by integrating the ghost image transmittance of at least one ghost image over the visible spectral band, and depends on the spectrum of the light source and the spectral light efficiency of the CIE 1964 photopic observer. The aforementioned transmission optical system has an ultraviolet protection index for the eye, expressed as E-SPF35°, The first surface includes a first coating, and the second surface includes a second coating. The aforementioned transmission optical system further includes at least another surface having another coating, Each pair of two different surfaces having the same curvature among the first surface, the second surface, and at least one other surface forms the same specific ghost image component having component ghost image transmittance. The first coating, the second coating, and the other coating are configured such that, for the same specific ghost image, the total ghost image transmission coefficient is calculated as a function of the sum of different components of the same specific ghost image transmittance and is lower than a predetermined threshold. The total ghost image transmission coefficient for each ghost image is given by the following formula: [Math 1] It is calculated using, Here, the visible spectral band extends from 380 to 780 nanometers. S(λ) represents the spectral brightness of the light source, [Math 2] represents the spectral efficiency of the CIE 1964 photopic observer, and ΣT(λ, 15°) represents the sum of all components of the ghost image transmittance for each pair of two surfaces having the same set of curvature. Transmission optical system.

2. A method for evaluating the visibility of a ghost image in a transmission optical system having at least a first surface and a second surface, The first surface and the second surface have different curvatures, and / or the first surface and the second surface are arranged to provide a refractive index. The method includes the step of determining colorimetric parameters for evaluating the visibility of a ghost image, The colorimetric parameters for the evaluation are based on the total ghost image transmission coefficient of at least one ghost image formed by the internal reflection of light rays from the light source between the first surface and the second surface and transmission through the transmission optical system. The light source is a light-emitting diode having a color temperature in the range of 2700 Kelvin to 6000 Kelvin. The light ray from the light source is incident on the first surface at a non-zero angle of incidence. The method for obtaining the total ghost image transmission coefficient is obtained by integrating the ghost image transmittance of at least one ghost image over the visible spectral band, and which depends on the spectrum of the light source and the spectral light efficiency of a CIE 1964 photopic observer.

3. A method for optimizing at least one anti-reflective coating of a transmission optical system having at least a first surface and a second surface, The first surface includes a first coating, and The second surface includes a second coating. At least one of the first coating and the second coating is an anti-reflective coating. The first surface and the second surface have different curvatures, and / or the first surface and the second surface are arranged to provide a refractive index. The aforementioned method, a) A step of determining the colorimetric parameters for evaluating the visibility of the ghost image, The colorimetric parameters for the evaluation are based on the total ghost image transmission coefficient of at least one ghost image formed by the internal reflection of light rays from the light source between the first surface and the second surface and transmission through the transmission optical system. The light source is a light-emitting diode having a color temperature in the range of 2700 Kelvin to 6000 Kelvin. The light ray from the light source is incident on the first surface at a non-zero angle of incidence. The total ghost image transmission coefficient is obtained by integrating the ghost image transmittance of at least one ghost image over the visible spectral band, and depends on the spectrum of the light source and the spectral light efficiency of a CIE 1964 photopic observer, and comprises a step, b) A step of modifying the structure and / or composition of the first coating and / or the second coating, c) Repeating steps a) and b) until the transmission optical system has a colorimetric parameter for evaluating the visibility of a ghost image that is lower than a predetermined threshold for the visibility of a ghost image, wherein the predetermined threshold for the visibility of a ghost image is less than 0.010%. Methods that include...

4. A system for evaluating the visibility of a ghost image in a transmission optical system having at least a first surface and a second surface, The first surface and the second surface have different curvatures, and / or the first surface and the second surface are arranged to provide a refractive index. The system for evaluating the visibility of the ghost image includes a processor configured to determine colorimetric parameters for evaluating the visibility of the ghost image, The colorimetric parameters for the evaluation are based on the total ghost image transmission coefficient of at least one ghost image formed by the internal reflection of light rays from a point light source between the first surface and the second surface and transmission through the transmission optical system. The light source is a light-emitting diode having a color temperature in the range of 2700 Kelvin to 6000 Kelvin. The light ray from the point light source is incident on the first surface at a non-zero angle of incidence. The total ghost image transmission coefficient is obtained by integrating the ghost image transmittance of at least one ghost image over the visible spectral band, and depends on the spectrum of the point light source and the spectral light efficiency of a CIE 1964 photopic observer.

5. The aforementioned transmission optical system has a third surface and / or a fourth surface, The system for evaluating the visibility of the ghost image includes a processor configured to determine the overall ghost image transmission coefficient for each ghost image formed by the internal reflection of the light ray between all pairs of surfaces having the same set of two different curvatures. The system for evaluating the visibility of a ghost image according to claim 4, wherein the total ghost image transmission coefficient depends on the sum of the components of the ghost image transmittances of all pairs of surfaces having the same set of two different curvatures.

6. The aforementioned transmission optical system has a third surface and / or a fourth surface, The system for evaluating the visibility of the ghost image includes a processor configured to determine the ghost image transmission coefficient of each ghost image formed by the internal reflection of light rays between each pair of surfaces having two different curvatures. The system for evaluating the visibility of a ghost image according to claim 4, wherein the total ghost image transmission coefficient depends on the sum of the ghost image transmission coefficients of each ghost image.

7. The method according to claim 2, The aforementioned transmission optical system has exactly two surfaces, The aforementioned ghost image is given by the following formula: T (λ, 15°) = T Cx (λ, 15°). R BCc (λ, α). R BCx (λ, α). T Cc (λ, α). (T int (λ, α)) 3 It has a ghost image transmittance calculated using, Here, λ represents the wavelength of the visible spectrum. The incident angle of the light source is set to 15 degrees and corresponds to the refraction angle α inside the substrate supporting the first surface and the second surface. T Cx (λ, 15°) represents the spectral transmittance of the first surface at the angle of incidence, R BCc (λ, α) represents the spectral reflectance of the second surface with respect to the reflection generated inside the substrate medium at the refraction angle α. R BCx (λ, α) represents the spectral reflectance of the first surface with respect to the reflection generated inside the substrate medium, T Cc (λ, α) represents the spectral transmittance of the second surface, and A method in which T int(λ, α) represents the spectral transmittance between the first surface and the second surface.

8. The total ghost image transmission coefficient is given by the following formula: [Math 3] It is calculated using, Here, the visible spectral band extends from 380 to 780 nanometers. S(λ) represents the spectral luminance of the light source, and [Math 4] The method according to claim 7, wherein is the spectral light efficiency of a CIE 1964 photopic observer.

9. The first surface comprises a first coating, and the second surface comprises a second coating. The aforementioned transmission optical system further includes at least another surface having another coating, Each pair of two different surfaces having the same curvature among the first surface, the second surface, and at least one other surface forms the same specific ghost image component having component ghost image transmittance. The method according to claim 2, wherein the first coating, the second coating, and the other coating are configured such that, for the same specific ghost image, the total ghost image transmission coefficient is calculated as a function of the sum of different components of the same specific ghost image transmittance and is lower than a predetermined threshold.

10. The method according to claim 2, wherein the predetermined threshold for the visibility of the ghost image is 0.007%.

11. The method according to claim 3, The aforementioned transmission optical system has exactly two surfaces, The aforementioned ghost image is given by the following formula: T (λ, 15°) = T Cx (λ, 15°). R BCc (λ, α). R BCx (λ, α). T Cc (λ, α). (T int (λ, α)) 3 It has a ghost image transmittance calculated using, Here, λ represents the wavelength of the visible spectrum. The incident angle of the light source is set to 15 degrees and corresponds to the refraction angle α inside the substrate supporting the first surface and the second surface. T Cx (λ, 15°) represents the spectral transmittance of the first surface at the angle of incidence, R BCc (λ, α) represents the spectral reflectance of the second surface with respect to the reflection generated inside the substrate medium at the refraction angle α. R BCx (λ, α) represents the spectral reflectance of the first surface with respect to the reflection generated inside the substrate medium, T Cc (λ, α) represents the spectral transmittance of the second surface, and A method in which T int(λ, α) represents the spectral transmittance between the first surface and the second surface.

12. The total ghost image transmission coefficient is given by the following formula: [Math 5] It is calculated using, Here, the visible spectral band extends from 380 to 780 nanometers. S(λ) represents the spectral luminance of the light source, and [Math 6] The method according to claim 11, wherein is the spectral light efficiency of a CIE 1964 photopic observer.

13. The first surface comprises a first coating, and the second surface comprises a second coating. The aforementioned transmission optical system further includes at least another surface having another coating, Each pair of two different surfaces having the same curvature among the first surface, the second surface, and at least one other surface forms the same specific ghost image component having component ghost image transmittance. The method according to claim 3, wherein the first coating, the second coating, and the other coating are configured such that, for the same specific ghost image, the total ghost image transmission coefficient is calculated as a function of the sum of different components of the same specific ghost image transmittance and is lower than a predetermined threshold.

14. The method according to claim 3, wherein the predetermined threshold for the visibility of the ghost image is 0.007%.

15. The system according to claim 4, The aforementioned transmission optical system has exactly two surfaces, The aforementioned ghost image is given by the following formula: T (λ, 15°) = T Cx (λ, 15°). R BCc (λ, α). R BCx (λ, α). T Cc (λ, α). (T int (λ, α)) 3 It has a ghost image transmittance calculated using, Here, λ represents the wavelength of the visible spectrum. The incident angle of the light source is set to 15 degrees and corresponds to the refraction angle α inside the substrate supporting the first surface and the second surface. T Cx (λ, 15°) represents the spectral transmittance of the first surface at the angle of incidence, R BCc (λ, α) represents the spectral reflectance of the second surface with respect to the reflection generated inside the substrate medium at the refraction angle α. R BCx (λ, α) represents the spectral reflectance of the first surface with respect to the reflection generated inside the substrate medium, T Cc (λ, α) represents the spectral transmittance of the second surface, and T int(λ, α) is a system representing the spectral transmittance between the first surface and the second surface.

16. The total ghost image transmission coefficient is given by the following formula: [Number 7] It is calculated using, Here, the visible spectral band extends from 380 to 780 nanometers. S(λ) represents the spectral luminance of the light source, and [Number 8] The system according to claim 15, wherein is the spectral light efficiency of a CIE 1964 photopic observer.

17. The first surface comprises a first coating, and the second surface comprises a second coating. The aforementioned transmission optical system further includes at least another surface having another coating, Each pair of two different surfaces having the same curvature among the first surface, the second surface, and at least one other surface forms the same specific ghost image component having component ghost image transmittance. The system according to claim 4, wherein the first coating, the second coating, and the other coating are configured such that, for the same specific ghost image, the total ghost image transmission coefficient is calculated as a function of the sum of different components of the same specific ghost image transmittance and is lower than a predetermined threshold.

18. The system according to claim 4, wherein the predetermined threshold for the visibility of the ghost image is 0.007%.

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