WO2020030701A1 - Method for designing a fresnel surface and fresnel surface obtained by such a method - Google Patents

Abstract

The present invention discloses a method for designing a Fresnel surface (104) which does not transmit unwanted artefacts such as stray light induced by reflection or refraction on draft facets (110) of the Fresnel surface (104). The invention also relates to an optical system obtained by such a method. The invention may be applied, but is not limited to Fresnel lenses (100) used in a Head-mounted display, such as a virtual or augmented reality device.

Description

METHOD FOR DESIGNING A FRESNEL SURFACE AND FRESNEL SURFACE

OBTAINED BY SUCH A METHOD

Description

Technical field

[0001] The invention relates to an optical lens and more particularly to a lens with Fresnel features for application in, but not limited to, a head-mounted display.

Prior art

[0002] The use of Fresnel lenses in a head-mounted display is known from the document WO 2018/052493 A1. Such a lens comprises alternated slope facets and draft facets and provides the compactness required in a head- mounted device. When using a Fresnel lens, optical artifacts occur, resulting from the reflection of light through the various facets of the Fresnel lens. In order to reduce these artifacts, the lens of the above-mentioned document comprises draft facets that are oriented such as to reduce the number of rays of light reaching the draft facets after refraction through a slope facet. In particular, this document uses draft facets that are orientated parallel to chief rays.

[0003] As will be detailed in the following paragraphs, such a device leaves room for improvement to reduce further the artifacts perceived by the user’s eye. Indeed, draft facets, peaks and valleys induce a decrease of contrast and resolution, and/or artefacts such as flare or ghost reflections in the image.

[0004] In prior art document WO 2018/052493 A1 , in order to minimize stray light generated by the draft facets, the main principle consists in the orientation of the draft facets along or nearby the chief ray angles (noted 0_£ and 0_t on figure 1 ) at the draft facet position given the two media from either side of the Fresnel surface. With this design and as explained further below, not all stray light is suppressed.

[0005] Figure 1 illustrates such a design of the draft facets. On the left side of figure 1 , a chief ray Rc propagates from the object to the image and is refracted by the slope surface 8. The chief ray angle is 9_t in medium n_P and 9_t in medium n_s. 0_£ and 6_t are related through n_F, n_s, 0_slope and Snell law.

[0006] On the left side of figure 1 , the draft surface 10 is oriented along the refracted chief ray angle 6_t. On the right side of figure 1 , the draft surface 10 is oriented along the incident chief ray angle 0_£.

[0007] An orientation of the draft surface along the incident chief ray angle 0_£ has the advantage of maximizing the light transmission of the groove. An orientation of the draft surface along the refracted chief ray angle 6_t has the advantage of minimizing the apparent size of the draft surface seen from the image side.

[0008] This main principle limits dramatically the range of draft surface angles at a given position of the Fresnel surface and therefore restricts substantially the flexibility of the design. Moreover, the maximalization of light transmission and the minimization of the apparent size of the draft surface are unnecessarily too severe constraints for the possible draft surface angles range: an imaging optical system can work properly without the maximalization of light transmission and the minimization of the apparent size of the draft surface. Furthermore, because imaging optical systems have finite size exit pupil or image, some rays nearby the chief ray angles will inevitably reach the draft surface. Part of those rays will have transmitted angles nearby the chief ray angle and consequently are highly susceptible to reach the exit pupil and generate stray light. Moreover, prior art considers only two kinds of stray light paths (inner total internal reflection and outer reflection) and it doesn’t consider the contributions of the whole object or entrance pupil for a considered imaging beam.

[0009] For a rotationally symmetric optical system, the chief ray along the optical axis crosses the Fresnel surface at his center and is normal to the Fresnel surface. At the center of the Fresnel lens, the angles are 0_slope = 0_t = 0_t = 0° . So, the prior art main principle leads to draft angles which tend towards zero, when the radial height on the Fresnel surface tends toward zero. [0010] Document US 2012/0120498 A1 discloses another example of a Fresnel lens. In this lens, the directions of the draft facets converge and intersect in the focal point, i.e. the eye of the user (see figure 3 of US 2012/0120498 A1 ).

[0011 ] Other documents show Fresnel lenses with a constant and small draft angle over the entire lens aiming at maximizing the light transmission.

[0012] All these lenses produce unwanted artefacts or stray light.

Summary of the invention

Technical problem

[0013] The problem solved by the present invention is to provide a lens with an improved design, such that fewer optical artifacts are transmitted through the lens.

Technical solution

[0014] The invention solves this problem with the appended independent claims.

Auxiliary features of the invention are encompassed by the dependent claims and the following description.

[0015] The solution to this stated problem is structured around three principles, all three aiming at removing unwanted phenomena when using a Fresnel lens.

[0016] The main and first principle consists in removing stray light by designing optical systems and their Fresnel surface(s) in a way which deviates or keeps out potential stray light rays from the exit pupil or image along the different possible stray light paths. The second principle aims at addressing more specifically the visible finite grooves size and to further avoid potential remaining stray light rays. The third principle consists in a further design work and optimization with or without the previous principles, led by manufacturing requirements or unreached lens requirements and applied in particular to unsharp peaks or valleys, grooves with too low transmittivity, draft, peaks or valleys surfaces including points inducing stray light. [0017] In the main and first principle, the propagation of a ray along a stray light path should end its path outside the perimeter of the exit pupil or image, or should be back propagated towards the object or entrance pupil side, or the incoming ray should not have an incident angle on the Fresnel surface which allows its propagation along the stray light path.

[0018] A geometrical formulation of this first principle is presented with the help of figures 2 and 3.

[0019] Let us consider an entrance pupil surface or object surface 3 with an arbitrary surface shape and an arbitrary perimeter shape.

[0020] Let us define a plane (j) orthogonal to the Fresnel surface at position EF, and a pair of points ( 0_+j ; 0__j ) located on the perimeter of the input pupil or object, such as incident rays at position EF coming from 0_+j and 0__j belong to the plane (j) as illustrated in figure 3.

[0021] The points which belongs to the perimeter of this object or entrance pupil can be expressed as a set of pairs of points, the rays of which are incident to EF and belong to a normal plane such as ( 0_+j ; 0__j ) pair of points and plane (j)·

[0022] Figure 2 illustrates a few examples of pairs of points located on the entrance pupil or object perimeter. Depending on the shape of the perimeter and on the object side optical system part (50 on figure 3), one can define several pairs of points per incident plane such as (0_+ml; 0__ml ) and (0_+m2,· 0__m2 for plane m.

[0023] Additional rays within the entrance pupil or image are incident range on Fresnel surface at position E_P within the same incident plane perpendicular to the Fresnel surface at position E_F.

[0024] For each pair of points, we define a segment of points in the entrance pupil or object formed by incident rays at position E_P belonging to the same plane as the pair of points. For example, the pair of points ( 0_+j ; 0__j ) defines the curve/segment [ 0_+j ; 0__j] represented in dotted line on figure 2.

[0025] For each pair of points, there exists a new pair of points representing segment or subsegment(s) of rays which propagates toward the image or exit pupil side for a given stray light path P_x. For instance, (0₊ ; O^p ) is the pair of points representing the segment of rays following stray light paths P_x which can propagate toward the image or exit pupil side. The points which do no propagate successfully result from incident rays which cannot fulfill the stray light path P_x propagation conditions or which finally propagates back toward the object or entrance pupil side.

[0026] With reference to figure 3, a Fresnel surface 1 is depicted, with an object or entrance pupil 30 on one side and an image or exit pupil 28 on the other side. Object side and image side optical elements 50, 50’ can be provided on one and/or the other respective side of the Fresnel lens. Depending on the object side optical system part 50, the relative position of the two rays incident at E_P is inverted or not. Their incident angles at position E_P are 0_im and 0_iM with 0_im < 0_iM.

[0027] Figure 3 shows a schematic representation of the targets of two rays coming from the pair of points (0^Px,· Oi^x) which reach the Fresnel surface at position E_P within the same incident plane (j), normal to the Fresnel surface at point £>. The plane (j) is defined by the intersection of the two incident rays n, r2 coming from the pair of points (0^Px,· Oi^x).

[0028] Considering infinitely small grooves, rays coming from the pair of point (0^Px,- Oi^x) are transmitted intersecting themselves and define the transmitted plane called ( ).

[0029] Let’s define 0'^Px, and 0'^Px, the location on the exit pupil or image side 28 of the rays coming from 0^Px and O ^x and following the stray light path P_x. Depending on the object side optical system part 50 and the stray light path, the relative position of the two rays transmitted on the exit pupil or image side is inverted or not.

[0030] We define the angles 0_tm and 0_tM with 0_tM > 0_tm which are defined by transmitted rays r₃ and r₄ belonging to the transmitted plane ( ). Rays r₃ and r₄ reach the image plane on two points /„, and / , of the perimeter of the exit pupil or image. [0031] Note: since j and j’ may not be coplanar, the measure of the angles is not made in accordance to the same line. Lines S_j and s\· illustrate this difference on figure 3.

[0032] The (curved) segment [/„,; / ,] is the intersection of rays with transmitted angle range from 9_tm to 9_m with the exit pupil or image.

[0033] In the particular case where exit pupil or image surface is flat, the segment [I_{M >} /¾,] may be a line segment.

[0034] Geometrical formulation of main principle: in order to avoid that stray light from the object reaches the exit pupil, all set(s) of pairs of points

located on the perimeter of, or inside of, the object or entrance pupil should, for each position E_P and each stray light path P_x, arrive outside of the image perimeter once propagated along the stray light path P_x.

[0035] For this purpose, one should ensure that the angle of transmitted rays in

,p_X ,rc

plane (/) coming from O .- and O

.- is outside the angle range [9_tm,· 9_m\. The geometrical formulation is illustrated in figure 3 in which the segment of rays \0'^r, 0'^Px ] on the exit pupil or image side is represented to arrive above or below the segment

In more complex optical lenses and/or exit pupils or image shapes, more than one range of angles is to be excluded within the plane (j’) such as [9_tml,· 9_tM1] and [9_tm2,· 9_{m 2}]

[0036] The geometrical formulation of this principle is illustrated on figure 3 wherein the segment of rays \0'^p_^xr, 0'^p ₊ ^x ] on the exit pupil or image side is projected above or below the segment

/ ,].

[0037] The main principle could be applied to pupil, object or image of any shapes including optical systems including separated areas of pupil, object or image surfaces. In that case, different segments should be considered within the same plane in the object or image side.

[0038] Mathematical formulation of the main principle for a rotationally symmetric optical system: in the specific case of a rotationally symmetric optical system with pupils, object and image without masked zones, for each stray light path P_x, it is sufficient to consider only one pair of points (O₊ ; 0__j ) and the radial positions of the draft E_F on the Fresnel surface. The center of the segment [o+y ; Oi*] is the center of the entrance pupil or object.

[0039] Rays r₃ and r₄ on figure 3 illustrate the limit rays and the range between these rays must be avoided. Two examples of rays (in dotted line on figure 3) above and below r₃ and r₄ respectively show examples of rays which do not transmit stray light to the exit pupil or image. Hence, for infinitely small grooves and rotationally symmetric optical systems, for a pair of points

there is no stray light if the output angles after the Fresnel surface deviation 9_t(9_im) and 9_t(9_m ) fulfill one of the two following conditions:

t (fi_lm) 9_tM and 9_t(9_iM) > 9_tM with 9_im and 9_m, coming from the points

), the minimum and maximum ray incident angles which fulfil the stray light path P_x propagation conditions (materialized by rays r₃ and r₄on figure 3).

[0040] The definition of output angles 9_t (9_im) and 9_t (9_iM) are provided below considering infinitely small Fresnel grooves, rotationally symmetric optical systems, a positive power transmissive Fresnel surface, n_s > n_F and a center of curvature oriented towards the image, for the following stray light path draft deviations: outer refraction, outer partial reflection, inner partial or total reflection, indirect inner draft refraction.

[0041 ] Outer refraction :

[0042] Outer partial reflection:

[0043] Inner partial or total reflection

[0044] Indirect inner draft refraction

with 0_siope the angle of the slope at the Fresnel surface position E_F .

[0045] Additional 9_t(9_im) and 9_t(9_iM) corresponding to more specific Fresnel surface definitions or corresponding to stray light paths depending of the rest of the optical system and intersecting the Fresnel surface one time or more should be computed as well where necessary.

[0046] Equivalent computation of output angles 9_t(9_im) and 9_t(9_m ) can be performed for any kind of system including non-rotationally symmetric imaging optical systems and any kind of Fresnel surface with positive or negative surface power, finite groove sizes, different type of surface (smooth, diffractive, holographic), optical indexes n_s and n_F, static or dynamic object, image, entrance and exit pupils optical surfaces coatings (reflective, transmissive, blackening) or surfaces treatment (partially or fully scattering) on draft or slopes or variable index materials.

[0047] In a rotationally symmetric optical imaging system, one may notice that with such a definition, when the radial height on the Fresnel surface tends toward zero, the draft angles doesn’t tend towards zero. For a rotationally symmetric optical system, the chief ray along the optical axis crosses the Fresnel surface at his center and is normal to the Fresnel surface. At the center of the Fresnel lens, the values of 9_tm and 9_m according to the invention are different from zero. Indeed, because total internal reflected rays and outer partially reflected rays should be deviated above or below the image or exit pupil, the main principle leads to draft angles which are different from zero, when the radial height on the Fresnel surface tends toward zero. [0048] Figure 4 provides a further illustration of a lens. In this case the draft angles are not aligned with chief rays but by following the rays of stray light, we observe that the equations expressed above are not fulfilled. This figure illustrates that foreseeing draft facets that are not along the chief rays is a necessary condition to our invention but is not sufficient. Rays R_ci and R_C2 are the chief rays reaching, through the slope facets 8, the center 20 of the exit pupil 28 for two different fields of view (two angular positions of the center 20). Rays R’_d, R’c₂ are parallel to rays R_ci, R_C2 to illustrate that both rays become stray light as a result of their interaction with draft facets 10. In particular, ray R’_d arrives directly on the draft facet and is divided into 2 components: the draft-refracted ray R’_{d a} and the outer draft-partially- reflected ray R’_{d b}. Ray R’_C2, after being refracted on a slope facet 8, experiments a draft inner partial reflection, as well as a draft refraction, both leading to a respective ray reaching the exit pupil. With the choice of draft angles made in this case, we observe that all stray-light rays arrive inside the exit pupil 28. As will be illustrated in an example below, by adequately choosing the draft angles, the stray-light could be sufficiently diverted to not intersect with the exit pupil 28.

[0049] The second principle that is of interest for our invention is to act on the fact that grooves of the Fresnel lens may become visible, as well as the peaks and valleys, i.e. the edging surfaces connecting draft facets to their neighboring slope facet or vice versa. This may occur for three reasons.

[0050] Firstly, the draft surfaces and peaks and valleys may become visible if their apparent size seen from the exit pupil is too big. Secondly, the draft surfaces and peaks and valleys are visible when the draft surface appears as a dim surface in the image. Thirdly, when the Fresnel lens shows an abrupt gradient between two successive draft facets, the difference may be perceived.

[0051] To those three causes correspond different artefacts in the image. The visibility of the groove depends in particular on the optical system, on the resolution of the optics and detection chain, and on the draft surface orientation. In particular, when the detector/receiver has a large accommodation distance and a large depth of field, which is the case of the human eye for instance, these three effects could appear. Each of these effects can be overcome by different actions. Firstly, by acting on the draft facet (through draft angle and/or draft facet groove depth, for instance), the apparent size of the draft facet can be reduced to a size small enough so that the peaks and valleys are not visible. Secondly, acting on the draft angle will reduce the diminishment of illumination (or dimness of the draft facet). Thirdly, a smooth variation between successive draft facets along the radius of the Fresnel lens solves the third phenomena. Of course, other ways exist in relation to the design of the optical system and its specifications.

[0052] The third principle that is of interest for our invention is to act on a stray light phenomenon by avoiding any draft facets, peaks or valleys along the path of the stray light. For this purpose, the number of grooves must be limited everywhere and especially in Fresnel surface regions where some stray light phenomena remain critical for the application. Moreover, peaks and valleys should ideally be perfectly sharp.

[0053] The third principle that is of interest for our invention is a design work and optimizations that can be done to further improve the transmission of the image while removing stray light.

[0054] An action to avoid any draft facets, peaks or valleys along the path of the stray light can be to limit the number of grooves especially in Fresnel surface regions where some stray light phenomena remain critical for the application.

[0055] An action on the sharpness of the facets and their edges is possible, for instance by using appropriate machining tools or procedures.

[0056] An appropriate choice or design trade-off of draft angles which does not reduce too much the transmission of light is advised.

[0057] The number of points of the surface which induce stray light could be reduced by the optimization of the shape of the surface. For example, a linear rotationally symmetric draft surface could be changed to an optimized curved rotationally symmetric draft surface in order to remove some critical points of the draft surface. [0058] A circumferentially non-homogenous design of the draft angles around the optical axis is possible.

[0059] The impact of surface points which induce stray light could be minimize or avoided by changing the surface definition at the points position. An action on the nature of the surface of the draft facet can be performed through coatings or treatments which absorb, reflect, deviate and/or spread away part of the remaining stray light. For example, some of these can be used: partially or fully scattering surfaces, blackened surfaces, diffractive or holographic surfaces, or transmissive or reflective optical coatings on the surfaces.

[0060] The less harmful stray light can be identified and the optical system can be designed to let it through.

[0061] The methodology to reduce the stray light is such that first should be checked that there is no possible adaptation in the optical system definition, the object, image, exit or entrance pupil size. Then, the first two principles should be applied to adapt the location and the definition of the considered Fresnel surface. Then finally the third principle is applied to define a trade- off so that the overall performance and stray light rejection of the imaging optical system can be optimized.

[0062] Each of the three principles listed above can be applied alone or in combination.

[0063] The present invention relates to a method for designing an optical system in accordance with claim 1. The invention also relates to an optical system, an optical element, a lens and a head-mounted display as defined in the appended claims. All of them aim at solving the same problem of removing stray light from being transmitted in an optical system.

[0064] In a particular embodiment, the invention relates to a lens comprising a substrate having a first surface and a second surface opposite the first surface, wherein the second surface comprises a plurality of radially alternated slope facets and draft facets, wherein the slope facets are arranged such as to make the rays of light coming from a source (or entrance pupil or object) to converge towards a zone (or exit pupil or image) where a receiver can be positioned, and wherein the draft facets are each oriented with respect to the optical axis of the lens by a respective draft angle, wherein the draft angles are such that: rays conning from said zone, refracting on the first surface, reflect inside the substrate on a draft facet and leave the lens through a slope facet along a direction that is such that they do not reach said source and/or rays coming from said source, reflecting on a draft facet before entering the substrate through a slope facet and leaving the lens through the first surface, leave the lens along a direction that is such that they do not reach said zone.

[0065] The substrate is substantially transparent to the human eye and/or to particular ranges of wavelengths. The lens may be axisymmetric around the optical axis or may be axisymmetric in at least a central portion of the lens. The lens can have any external shape and is not limited to a ring shape. The lens has an optical axis defined by the geometrical axis of the ray of light the direction of which would remain unchanged when passing through the lens.

[0066] The slope and draft facets build a Fresnel pattern. The slope facets are arranged (in position, orientation, shape of the surface, etc.) such that they play the role altogether that would play a non-Fresnel lens (convergent or not).

[0067] The source may be a two-dimensional plane or curved display, a three- dimensional scenery, etc. The source may be the physical entity representing the concept mentioned above as the entrance pupil or object. The zone may be the physical volume representing the concept mentioned above as the exit pupil or image.

[0068] Thus, by comparison to known lenses, the present invention deviates the rays on the draft facets, instead of reducing the rays that reach the draft facet for a given imaging path. The draft facets are however oriented such as to deviate the light outside of the zone of convergence of the slope facets. Flence, for a head-mounted display, such a lens makes it possible, among others, to avoid any stray light from reaching the ocular pupil after a total internal reflection or partial reflection on the draft facets. [0069] According to preferred embodiments, the lens of the invention can comprise one or more of the following features in any possible combination:

- the apparent size of the draft facet can be reduced to a size small enough so that the peaks and valleys are not visible, namely for draft facets at a distance from the optical axis greater than a predetermined threshold, the draft angle is of about 30° or less;

- the lens is such that for draft facets at a distance from the optical axis greater than a predetermined threshold, the depth of the draft facet is of about 300 pm or less;

- for draft facets at a distance from the optical axis greater than a predetermined threshold, the draft angle is comprised between 18° and 30°;

- at least two of the embodiments above are combined;

- at least one of the predetermined thresholds is comprised between 50% and 80% of the radius of the lens. It can be any integer % between these values;

- at least two radially consecutive draft facets have the same draft angle and preferably all the facets have the same draft angle, or do not vary of more than 10% over the entire lens. The draft angle of said parallel facets can be of about 35°, about 36° or about 38°;

- all of the draft angles are greater than 18°;

- the diminishment of illumination resulting from the orientation of the draft facet is such that the groove is not visible, for instance the diminishment is of less than 30%;

- the variation of draft angle between two successive draft facets is smooth, preferably below 20° angle, and most preferably below 18°;

- the connecting surface between a draft facet and its neighboring slope facet has a bending radius of less than 5 pm, preferably less than 2 pm. The theoretical edge between two neighboring facets is indeed in practice not precisely an edge but a connecting surface. This surface should however be as small as possible to avoid stray light; - the value of the draft angle of the draft facet at the optical axis, if any, is different from 0°. Indeed, by opposition to a lens where the draft facets are parallel to chief-rays, and thus wherein the draft facet at the center would be parallel to the optical axis, the draft facet at the center in the lens of the present invention is not at 0° angle;

- the draft angles of the at least three draft facets that are closest to the optical axis are higher than 20°, preferably higher than 28°;

- none of the draft facets is parallel to a chief-ray. Hence, none of the draft facets is oriented according to the direction of a ray of light coming from the zone after being refracted by the substrate;

- the zone is a volume centered on the optical axis of the lens at a distance comprised between 2 and 10 centimeters from the first surface and confined within a diameter comprised between 2 and 10 millimeters around the optical axis. This zone is expected to cover all possible positions of the pupil in various field of views and for various anatomy of users;

- the zone is a volume centered on the optical axis of the lens at a distance comprised between 3 and 20 times the average thickness of the lens and confined within a diameter around the optical axis that is comprised between 5% and 30% of the external diameter of the lens;

- the lens has a size between 2 and 15 centimeters in width and height, i.e. the two directions perpendicular to the optical axis and perpendicular to each other;

- the receiver is the pupil of a human eye;

- the source is a two-dimensional source arranged perpendicularly to the optical axis. A flat or curved display can form part of the source. Alternatively, the source can extend in three dimensions. It can be a display with additional optical elements, and/or a three-dimension virtual or real-life scenery;

- the source is at a distance from the second surface comprised between 5 and 50 mm; - the source has a dimension in the direction perpendicular to the optical axis that is comprised between 50% and 150%, of the dimension of the lens along the same direction;

- both the slope facets and the draft facets are substantially conical surfaces;

- the first surface is a continuous surface. It is preferably uniformly curved or flat;

- the first surface also comprises a series of alternated slope and draft facets. In such a case, any of the properties or orientation of the slope or draft facets of the second surface described above can be adapted to the slope or draft facets of the first surface;

- the substrate is made of germanium and/or the first and second surfaces are coated with germanium.

[0070] In the context of the present invention, a partial reflection occurs when the ray reflects partially on a draft facet outside of the substrate. The ray reflects partially before entering the substrate through a slope facet and refract partially through the draft facet. Both part-rays exit the substrate through the first surface.

[0071] In the context of the present invention, a total internal reflection occurs when a ray entering the substrate through the first surface reflects internally totally on a draft facet of the second surface before exiting the substrate through a slope facet.

[0072] The invention also relates to the use of a lens as described above in an imagery device, wherein the receiver is an image sensor and the source is an environment that is to be captured.

[0073] The image sensor can be of any appropriate type, including but not limited to CCD detector, infrared detector, etc.... The environment to be captured can be any two-dimensional image or three-dimensional scenery.

Advantages of the invention

[0074] By diverting the rays off the zone where the user eye can be, the artifacts due to stray light are totally removed from the field of view of the user, irrespective of the user’s morphology or the position of his/her eyes (hence field of view).

[0075] Also, the grooves that could become visible with other choices of draft angles are removed.

[0076] By the single action on the angles of the draft facets, four different kinds of stray light can be removed at once.

Brief description of the drawings

[0077] Figure 1 shows a cross section of two examples of known draft and slope facets;

[0078] Figure 2 shows an object of arbitrary shape;

[0079] Figure 3 depicts the stray light rays from object to image through point EF of the Fresnel surface;

[0080] Figure 4 shows a cross section of a known lens;

[0081] Figure 5 illustrates top view of a known the lens;

[0082] Figures 6 and 7 show the paths of geometric rays reflecting totally internally on the draft facets and originating from a zone, in the respective case of the lens of figure 5 and the lens according to the invention;

[0083] Figure 8 shows the irradiance in the plane of the display with the respective lenses of figures 6 and 7;

[0084] Figures 9A and 9B show a comparison of the known lens (figure 9A) and the lens of the invention (figure 9B) for stray light arising from total internal reflection;

[0085] Figures 10 and 11 show the paths of geometric rays reflecting partially on the draft facets of the lenses of figures 6 and 7, respectively;

[0086] Figure 12 shows the irradiance in the pupil plane with the respective lenses of figures 6 and 7;

[0087] Figures 13A and 13B show a comparison of the known lens (figure 13A) and the lens of the invention (figure 13B) for stray light arising from outer partial reflection; [0088] Figure 14 shows the paths of geometric rays directly refracting on the draft facets of the lens according to the invention;

[0089] Figure 15 shows the paths of geometric rays indirectly refracting on the draft facets of the lens according to the invention;

[0090] Figure 16 shows a graphical analysis of the draft angles which can be used in accordance with one aspect of the invention;

[0091 ] Figure 17 illustrates the visibility of the grooves;

[0092] Figure 18 illustrates the flare occurring with non-sharp peaks and valleys;

[0093] Figure 19 illustrates a head-mounted device according to the invention.

[0094] Figure 20 discloses a flowchart of the method according to the invention;

Description of the preferred embodiments

[0095] While the previous paragraphs detail the main issues that are encountered, the main concepts of the invention and an exemplary mathematical model for realizing the invention, the following describes the graphical results that are obtained with a Fresnel surface designed in accordance with the invention in comparison to the prior art lens, in the particular case of a head- mounted display, which is by no means limiting the invention.

[0096] As a general remark, it has to be noted that the facets of the Fresnel lens shown on the drawings are not drawn at scale and only a few of them are shown in the aim of illustrating the paths of the rays.

[0097] The axial direction is along the optical axis of the lens, the radial direction is perpendicular to the axis. The width of the lens is its axial dimension. The diameter, radius, width or height of the lens or of a facet of the Fresnel surface is measured radially. The depth of a groove formed by two adjacent facets of the Fresnel surface is measured axially.

[0098] Moreover, the illustrated rays are only geometric and theoretical rays used to illustrate the design of the lens and the directions taken by a ray. Although some drawings or the present description sometimes describe a ray as originating from the eye, such a ray is actually a geometric beam and not a ray of light. This is done for facilitating the understanding of the construction of the lens.

[0099] The invention concerns any kind of visible and non-visible wavelengths imaging optical systems including Fresnel surface(s) such as eyepiece optics (for example: head-up display optics, head-mounted display optics, augmented-reality optics, mixed-reality optics, or near-to-eye display optics), imaging lenses, relay optics or image-projection optics.

[00100] The principle of the invention is valid on any kind of Fresnel surface; rotationally symmetric or not, with a flat, spherical, aspherical or free-form slope definition, transmissive or reflective, positive or negative power. The optical index on each side of the Fresnel surface could be solid, liquid or gaseous, with an opposite surface or within a prismatic component. The grooves could have any kind of orientation such as circular, elliptic, linear or free-form within a plane, curved, aspherical or free-form surface. The slope and draft facets themselves could be flat, curved, free-form, diffractive or holographic.

[00101 ] This invention could be associated or optimized in combination with other Fresnel surface definition parameters which have an impact on stray light or other optical system constraints, requirements, performances such as groove width, groove depth definitions, peaks and valleys shape, optical surfaces coatings (reflective and/or transmissive) or draft surface treatment (draft black, mirror and/or draft partially or fully scattering surface).

[00102] The invention shows also to be useful for non-imaging inventions using Fresnel surface(s) such as light source beam shaping or beam focusing for which we need to have an image illumination without stray light.

[00103] The principles of the invention could be extended to other kind of surfaces or volumes which generate stray light in imaging system such as diffractive surface, hybrid diffractive-refractive surfaces, spatial light modulators.

[00104] Figure 5 shows a top view of a lens 1 where the draft facets are oriented as on left part of figure 1 , i.e. along chief rays. The lens has a first surface 2 and a second surface 4, opposite the first surface 2. These surfaces 2, 4 define the spatial limits of a substrate 6. Only part of the lens 1 is shown. [00105] The second surface 4 comprises a plurality of alternated slope facets 8 and draft facets 10. The slope facets 8 play the role of a regular lens, but by providing a discontinuous surface 4, the lens is thinner than an equivalent regular lens. The draft facets 10 connect the slope facets 8.

[00106] The lens 1 has an optical axis 12 defined by a central line parallel to a ray of light which would not be deviated by the lens.

[00107] The eye 20 of a user can be positioned along the optical axis 12. The eye has a pupil 22 which can take various positions in rotation around the center 24 of the eye 20. When focusing on different field of views, the pupil 22 rotates. The total field of view is shown by line 26 representing the nose- side extreme position of the pupil 22 and line 27 representing the temporal- side extreme position of the pupil 22. The pupil 22 can take various positions inside a zone schematically drawn with dotted lines as 28. This zone 28 can be approximated by a cylinder centered on the optical axis 12 and hence can be defined for example by a range of distance to the lens 1 and a diameter. In practice, this zone 28 corresponds to the exit pupil of the examples given above, i.e. defined only by a diameter and a distance to the lens.

[00108] Figure 5 shows two examples of geometrical rays R1 , R2 originating from the center of the pupil 22 for two positions of the pupil 22. The rays R1 , R2 are refracted when passing through the first surface 2 into chief rays RT, R2’. By design, the lens of the prior art is such that the draft facets are orientated along the chief-rays. The rays leave the lens as R1” and R2”.

[00109] When a source 30 such as a display is positioned on the side of the lens 1 opposite the eye 20, such a design of the draft facets 10 results in some stray light being sent from the display 30 to the eye 20. This is not desired, since only the light coming from the display 30 through the slope facets 8 is purposefully transmitted to the eye to form the image on the retina.

[00110] Figure 6 shows the lens of figure 5 with a bundle of rays R, R’, R” for all the draft facets 10 and originating from the zone 28. An optical element 50 is optionally positioned between the lens 1 and the display 30. The optical element can be a second positive power optical component such as a lens, or a second Fresnel lens with its Fresnel surface oriented towards the exit pupil or the image.

[001 1 1 ] As in figure 5, the light rays R” coming from the display and interacting with draft facets enter in the substrate 6 with a direction parallel to the draft facet and reach the zone 28 of the eye 20. Flence, we can observe that hundreds of unwanted rays reach the zone 28 of the eye 20. Rays entering the slope facets 8 and reflecting totally internally on the draft facets 10 also reach this zone 28.

[001 12] Figure 7 shows the same view but with a lens 100 according to the invention instead of lens 1 . The referral numbers are unchanged for the parts that are already discussed in previous figures and which remain the same. The referral numbers in relation to the lens of the invention 100 are incremented by 100 compared to the lens 1 known from the prior art. In this particular example, a constant draft angle Q of 36° has been used.

[001 13] It is important to note that the lens 100 can be flat or curved. The surface opposite to the Fresnel surface could be flat or non-flat. The focal length of lens 100 is larger than the focal length of the optional additional element 50.

[001 14] An exemplary embodiment of the lens 100 is a lens of a diameter of about 10 cm with a total field of view of about 140°. The element 50 can be a Fresnel surface with its grooves oriented towards the exit pupil. The focal of the first lens 100 varies between 30 mm and 75 mm. The focal length of the second lens 50 varies between 55 mm and 300 mm. The display has a diagonal size between 50 mm and 80 mm.

[001 15] The bundle of geometrical rays drawn from the zone 28 of the eye 20 refract on the first surface 102 and then reflect totally internally on the draft facets 1 10 to exit the lens through slope facets 108. Their orientation at the exit of the lens 100 is such that (whether or not the optical element 50 is present), none of the rays R” reaches the display 30. In practice, the display 30 is mounted in a dark environment of a head-mounted display. Flence, the display 30 is the only source of light. This means that none of the geometrical rays R” which interacts with the draft facets 1 10 and which passes through the zone 28 contains any light. Hence, there is no stray light transmitted through these rays.

[001 16] A zoomed-in portion of figure 7 shows the total internal reflection of a ray on the draft facet 1 10.

[001 17] Figure 8 shows the irradiance of the display plane with the rays shown on figure 6 (dotted line) and on figure 7.

[001 18] We can see here the substantial differences between the known lens 1 and the lens 100 according to the invention. With the known lens 1 , a substantial number of rays from the zone 28 irradiate the display, whereas with the lens 100, the irradiance is null over the entirety of the display width/height.

[001 19] Figures 9A and 9B illustrate the comparison of the prior art lens and the lens of the invention for four different field of views. For each field of view (or pivot angle of the ocular pupil), a square stimulus has been centered at the position on the display of the chief ray associated to each fixation direction. From a perceptual point of view, an important relevant phenomenon is those affecting stimulus close to the gaze direction. The pictures show the simulated images on the retina of the square source employed plus the stray light arising from one total internal reflection on the draft facets of the Fresnel surface. As can be seen on fig. 9A with the lens of prior art, this stray light gives rise to inner-flare tails, directed towards the axis of the optical system. The higher the FOV, the higher the flare irradiance. In contrast, with the lens according to the invention (on figure 9B), there is no flare arising from total internal reflection on the draft facets.

[00120] Figure 10 illustrates a bundle of rays initiated by the display 30 and reflecting partially on the draft facets 10 before penetrating the substrate through the (adjacent) slope facets 8. All the rays converge towards the zone 28.

[00121 ] By contrast, the orientation of the draft facets of the lens 100 on figure 1 1 are such that the rays initiated by the display 30 and reflecting partially on the draft facets 1 10 before penetrating the substrate through the (adjacent) slope facets 108 do not converge towards the zone 28 but rather diverge outside of the zone 28. [00122] A zoomed-in portion of figure 1 1 shows the partial reflection of a ray on the draft facet 1 10.

[00123] Figure 12 shows the irradiance on the pupil plane with the rays shown on figure 10 (dotted line) and on figure 1 1 .

[00124] We can see here the substantial difference between the known lens 1 and the lens 100 according to the invention. With the known lens 1 , all the rays from the display 30 reflecting partially on the draft facets 10 irradiate the pupil 22, whereas with the lens 100, none of the rays reflecting partially on the draft facets 1 10 penetrates the zone 28. Hence, with the lens 100, the only rays from the display 30 which reach the eye are those passing through the slope facets 108 without interacting with the draft facets 1 10.

[00125] Figures 13A and 13B show a comparison similar to the one made on figures 9A and 9B, between the lens of prior art and the lens of the invention. In this case the outer partial reflection generates outer flare for a lens according to prior art (figure 13A). The lens according to the invention (figure 13B) is free from flare.

[00126] Figure 14 shows the lens of the invention with rays of direct refraction. The draft angles are such that direct refraction on the draft facets is diverted away from zone 28. One can easily imagine that if the draft angles would be too high, the light rays directly refracting on the draft facets would propagate towards zone 28. A zoomed-in section of figure 14 illustrates a light ray’s path.

[00127] Figure 15 shows the lens of the invention with rays of indirect refraction.

Indirect refraction refers to ray of light refracting on the draft facet after exiting the lens through a slope facet. The draft angles are such that indirect refraction on the draft facets is diverted away from the source 30. A zoomed- in section of figure 15 illustrates a light ray’s path.

[00128] In relation to figure 16, an example is given for ranges of optimal draft angles Q. Needless to say, all values are here dependent on many parameters and the invention is not limited to these values of draft angle. Also, and as explained above in details, the skilled person would know how to calculate and define the optimal ranges depending on the parameters he uses. The optimal draft angles may vary substantially if some parts of the optical system are changed.

[00129] Using the method explained before for the first principle exposed above and its mathematical formulation of the stray light paths P_x, the draft angles are to be determined after the other parameters are set. The draft angle is therefore the only degree of freedom. For each stray light path and for each distance to the center of the lens (or radial height of the Fresnel surface), it is thus possible to study the stray light and the draft angles which correspond to the two rays starting from O₊ and Oi * and intersect the exit pupil at / ,

Then, using the inequality conditions, we determine the preferred range of draft angles which do not generate stray light. We can also exclude draft angles ranges which are not manufacturable (negative angles are non-manufacturable undercuts).

[00130] Figure 16 shows the draft angles (in degrees) in function of the radial height of the facet (distance to the optical axis).

[00131 ] The draft-angle distribution proposals for different radial heights on a Fresnel surface are shown, which are able to take the different stray-light paths away from the pupil in different ways. Curves with black-filled markers correspond to the outer partial reflection, while the white-filled markers are for the inner partial or total reflection. Triangles designate the cases where stray light coming from the O₊ point on the display margin are deviated to the I^_j, pupil border, while squares indicate that light from O * on the display are taken to the /„, pupil border.

[00132] The line with the black circles represents the limit for the choice of the draft angle due to direct refraction on the draft facets: when the draft angle is too high, light refracts directly on the draft facets and reaches the exit pupil. This higher limit is merely restrictive for facets that are close to the optical axis. An illustration of adequately chosen draft facets for direct refraction is shown on figure 14 discussed above. Direct refraction on the draft facets is not an issue for lenses according to prior art, since a draft facet oriented along chief ray will not refract by direct refraction towards the exit pupil. [00133] The line with the stars illustrates the limit line above which the grooves become visible (second principle). We refer to figures 17 and 18 with this respect.

[00134] Further draft angles distributions are shown in the graph: a continuous line on top of the graph shows a draft angle almost constant (between 34 and 36°) which succeeds in deviating all stray light from these phenomena away from the exit pupil for all radial heights on the Fresnel surface; and a dashed line curve shows the distribution of draft angles according to prior art, where the draft angle follows the direction of the chief ray inside the substrate medium (traced from a pupil located on the ocular center of rotation). The latter curve is graphically situated between the square and triangle markers and fails therefore to eliminate stray light over the entire radius. At the central radial region, it takes none of the phenomena outside of the pupil.

[00135] In conclusion, according to the first principle, the ranges of draft angles which fulfils the need to prevent stray light are above a threshold of 30° at low radius, this threshold slowly increasing with the radius. Also, another range at high radius (about two thirds of the total radius of the lens) is free from stray light. Both ranges are illustrated on figure 16 with hatching.

[00136] As noted above, the theoretical range below 0° is excluded for manufacturing reasons.

[00137] By applying the second principle, one would note that with the optical system used for figure 16, one should limit the groove to about 300 pm in the outer region of the Fresnel surface. In the inner region, higher values can be used. For this reason, draft angles should be limited to about 40° especially in the outer region of the Fresnel surface. Depending on the parameters of the optical system, this limit could go down to 30°. In the inner region, higher values can be used.

[00138] Flence, if both first and second principles are to be respected, the combination of their teaching leads to a distribution of draft angles that is high (above 30°) for about two thirds of the facets, closer to the optical axis, and draft angles between 18 and 30° for draft facets in the last radial third of the lens. Of course, the threshold of two thirds is here only given as example and by varying the input data (size of the lens, object, exit pupil, distance between them, etc.), other values of optimal ranges for the draft angle and other threshold would be calculated.

[00139] According to the third principle, the answer may consist in reducing the number of grooves, especially in the radially outer region of the lens. For example, in the radially outer third of the lens, the maximum groove size (width and/or depth) of about 300 pm could be used. In the inner region of the Fresnel surface, lower or higher values could be used.

[00140] Also, the edges (peaks or valleys) connecting two adjacent facets (draft and slope facet) are machined so as to build a radius that is of 2 pm or less.

[00141 ] On figure 16, the thresholds S2 and S3 corresponding to the respective second and third principle are indicated at about the same positions. These thresholds can however be different, depending on multiple factors inherent to the optical system.

[00142] In conclusion for this specific situation, for half field of view lower than 40°, a draft angle value between 15° and 30° should be chosen and for fields of view higher than 40°, draft angle values between 15 ° and 40° can be chosen.

[00143] With the present definition of the optical system and Fresnel surface, the groove width should be limited to reach a frequency between 3 and 4 cycles per mm in the outer region of the Fresnel surface. In the inner region, lower values can be used.

[00144] With the present definition of the optical system and Fresnel surface, the draft angle should be limited to about 35° or about 36° especially in the outer region of the Fresnel surface. In the inner region, higher values can be used.

[00145] The peaks and valleys should be surfaces with a size of 5 pm or less.

[00146] Exemplary ranges of values for the various parameters are given in the following paragraphs. Parameters such as the focal length, the field of view of the lens, the curvature of the substrate, etc., impact the ranges of values presented as optimal here for a specific set of parameters. [00147] The following are two examples of design which fits, for specific conditions, the requirements of the principles exposed above.

[00148] In the first example, the draft angles are given relatively to the chief-rays orientation. Example 1 : between 0° and 20° of half field of view, the draft orientation are between 10° and 30° superior to the draft orientation along chief ray within Fresnel surface substrate; between 50° and 70° of half field of view, the draft orientation are between 1 ° and 20° superior or between - 1 ° and - 20° inferior to the draft orientation along chief ray within Fresnel surface substrate; in the mid-range between 20° and 50°, the draft orientation is in a range which evolves smoothly from the two previously- mentioned ranges, or in other words the allowed ranged is a convergence of the lower and higher ranges.

[00149] In the second example, the draft angles are given in absolute values.

Example 2: between 0° and 20° of half field of view, the draft orientation should be between 15° and 35°; between 50° and 70° of half field of view, the draft orientation should be between 25° and 35° or between 15° and 30°; in the mid-range between 20° and 50°, the draft orientation is in a range which evolves smoothly from the two previously-mentioned ranges, or in other words the allowed ranged is a convergence of the lower and higher ranges.

[00150] In both examples, the groove frequency between 0° and 20° could be between 0.05 cycles/mm and 4 cycles/mm and could be between 0.5 cycles/mm and 6 cycles/mm between 50° and 70°.

[00151 ] On figure 17, the dashed line shows the groove frequency (number of grooves per mm) below which the grooves are large enough to be noticed, in function of the relative distance to the optical axis. The full line shows the selected groove frequencies in the particular example of the invention, which allows to maximize the groove width at higher radial heights, without the grooves being noticeable.

[00152] Still in relation to the third principle, figure 18 depicts the difference between the present invention (top graph) with ideally sharp peaks and valleys, where one notices the absence of stray light and the apparition of inner and outer flare with the 15 pm curved valleys (bottom graph). [00153] Figure 19 shows schematically in top view a head-mounted device 200. The device comprises two displays 30, two respective lenses 100 for illuminating two respective zones 28 arranged to receive the pupils 22 of both eyes 20 of a user. The optical axes 1 12 of the two lenses 100 intersect at an angle higher than 30°.

[00154] Figure 20 discloses a flowchart of a particular method according to the invention. In this very specific example, all the parameters are fixed and the only degree of freedom remaining is the draft angle for each of the draft facets. The method comprises three main steps 1000, 2000, 3000, to be performed in that order.

[00155] In a first step 1000, the dimensions and relative positions of the elements are determined: the position of the optical axis 1 12, the size and position of the zone 28, the size and position of the display 30 and of the optional optical element 50, and the position and dimensions of lens 100.

[00156] In a second step 2000, the slope facets 108 of the lens 100 are determined: their number, their dimensions and positions, radially, axially and in orientation are defined. The slope facets 108 are such that the image of the display converges towards the zone 28, and such that the thickness of the lens is as desired.

[00157] In a third step 3000, the draft angles Q of the draft facets 1 10 linking two successive slope facets 108 are determined. At the beginning of the third step 3000, the only“degree of freedom” left for the system is the choice of the draft angles. These are determined as presented above by choosing appropriate angles which fit the above-mentioned equations. Flence, the draft angles are such that no stray light reaches the zone 28, be it by total internal reflection or partial reflection. The third step may be carried out by appropriate computer-aided design software or by iterations when searching for a suitable range of angles for each facet. A minimum or a constant angle can also be sought for, as presented below with an angle of 36°, after which a variation or optimization process of such an angle can be carried out. [00158] It is important to note that although the present invention has been described in the context of a head-mounted display, the invention is not limited to this area since diverting the stray light produced by unwanted phenomena in a system using a Fresnel surface can show to be useful in many different technical fields.

[00159] The skilled person would also recognize that all the embodiments described above can be combined insofar as they do not contradict each other.

[00160] As illustrated on figure 2, the source or object that is to be projected onto the exit pupil can have any size, shape or form. It can be at any distance to the lens.

[00161 ] Although it may seem that the invention focuses on the choice of the draft angles, the principles exposed above can be applied to any parameter of the optical system, including but not limited to, the relative position of the various elements of the system, the power of the lens, the geometry of the slope facets, the index of refraction of the media, the wavelength, etc.

Claims

Claims

1. Method for designing an optical system comprising a Fresnel surface (104), the method comprising a step of determination of parameters of the optical system, such as the geometry of the Fresnel surface (104), the geometry of an entrance pupil or object (30), the geometry of an exit pupil or image (28), the relative distance between the pupils and the Fresnel surface (104), such that for each pair of points ( 0_+j ; 0__j ) of the entrance pupil (30) defined by the intersection of the entrance pupil (30) with a plane (j), and for each point (EF) of the Fresnel surface (104), at least one of the following pairs of inequations is verified:

^t^fiirn tm 3nd 0 (^£M) tm

for at least one of the four, and preferably all, of the following pairs of equations:

with 0_S]Ope the angle of the slope facet at the Fresnel surface position E_F, Q _draf_t the angle of the draft facet at the Fresnel surface position E_F, 9_im and 9_m the incident angles from the pair of points at position E_F with im iM

9_t(9_im) and 9_t(9_iM) the transmitted angles at position E_P,

9_tm and 9_m the extreme transmitted angles at position E_P, reaching the extreme end points (/ ,; / , ) of the exit pupil (28), and n_P and n_s the indices of refraction of the mediums upstream and downstream point E_P .

2. Optical system obtainable by the method of claim 1.

3. Optical element (100) comprising a Fresnel surface made of a plurality of alternated slope facets (108) and draft facets (110), wherein the geometry of the slope facets (108) and draft facets (110), the relative distance to the Fresnel surface and the geometry of the entrance pupil (30) and of the exit pupil (28), and the index of refraction of the media on both sides of the Fresnel surface are such that it can form, with the entrance pupil and the exit pupil, an optical system according to claim 2.

4. Lens comprising a Fresnel surface, the lens being preferably an element according to claim 3, wherein:

rays coming from the entrance pupil (30), refracting on a draft facet (110) exit the Fresnel surface along a direction that is such that they do not reach the exit pupil (28);

rays coming from the entrance pupil (30), reflecting on a draft facet (110) towards a slope facet (108) and exit the Fresnel surface along a direction that is such that they do not reach the exit pupil (28);

rays coming from the exit pupil (28), reflecting on a draft facet (110) before going through a slope facet (108), leave the lens (100) along a direction that is such that they do not reach the entrance pupil (30) and/or

rays coming from the exit pupil (28), refracting on a slope facet (108) before exiting the lens and refracting on a draft facet (110), go through a slope facet (108) before leaving the lens (100) along a direction that is such that they do not reach the entrance pupil (30).

5. Lens (100) preferably according to claim 4 comprising a substrate (106) having a first surface (102) and a second surface (104) opposite the first surface (102), wherein the second surface (104) comprises a plurality of radially alternated slope facets (108) and draft facets (1 10), wherein the slope facets (108) are arranged such as to make the rays of light coming from an entrance pupil (30) to converge towards an exit pupil (28) where a receiver (22) can be positioned, and wherein the draft facets (1 10) are each oriented with respect to the optical axis (1 12) of the lens (100) by a respective draft angle (Q), characterized in that the draft angles (Q) are such that: rays coming from said exit pupil (28), refracting on the first surface (102), reflect inside the substrate (106) on a draft facet (1 10) and leave the lens (100) through a slope facet (108) along a direction that is such that they do not reach said entrance pupil (30) and/or rays coming from said entrance pupil (30), reflecting on a draft facet (1 10) before entering the substrate through a slope facet (108) and leaving the lens (100) through the first surface (102), leave the lens (100) along a direction that is such that they do not reach said exit pupil (28).

6. Lens (100) according to any of claims 4 or 5, characterized in that the draft angles (Q) are such that the apparent size, in width and/or depth, of the draft facet can be reduced to a size small enough so that the peaks and valleys are not visible, namely for draft facets (1 10) at a distance from the optical axis (1 12) greater than a predetermined threshold (S2, S3), the draft angle (Q) is of about 38° or less, preferably about 35° or 36° or less.

7. Lens (100) according to any of claims 4 to 6, characterized in that the draft angles (Q) are such that for draft facets (1 10) at a distance from the optical axis (1 12) greater than a predetermined threshold (S2, S3), the depth of the draft facet is of about 300 pm or less.

8. Lens (100) according to any of claims 4 to 7, characterized in that the draft angles (Q) are such that for draft facets (1 10) at a distance from the optical axis (1 12) greater than a predetermined threshold (S2, S3), the draft angle (Q) is comprised between 18° and 30°.

9. Lens according to any of the claims 4 to 8, characterized in that at least one of the predetermined thresholds (S2, S3) is comprised between 50% and 80% of the radius of the lens.

10. Lens (100) according to any of claims 4 to 9 characterized in that none of the draft angle (Q) is equal to 0°.

1 1 . Lens (100) according to any of claims 4 to 10, characterized in that all of the draft angles (Q) are greater than 18°.

12. Lens (100) according to any of claims 4 or 1 1 , characterized in that at least two radially consecutive draft facets (1 10) have the same draft angle (Q) and preferably all the draft facets (1 10) have the same draft angle (Q) or do not vary of more than 10% over the entire lens.

13. Lens according to claim 12, characterized in that the draft angle (Q) of the facets is about 35°, about 36° or about 38°.

14. Lens (100) according to any of claims 4 or 13, characterized in that the diminishment of illumination resulting from the orientation of the draft facet is such that the groove is not visible, for instance the diminishment is of less than 30%.

15. Lens (100) according to any of claims 4 or 14, characterized in that the variation of draft angle (Q) between two successive draft facets is smooth, preferably below 20° angle, and most preferably below 18°.

16. Lens according to any of claims 4 to 15, characterized in that the connecting surface between a draft facet and its neighboring slope facet has a bending radius of less than 5 pm, preferably less than 2 pm.

17. Lens (100) according to any of claims 4 to 16, characterized in that the draft angles (Q) of the at least three draft facets (1 10) that are closest to the optical axis (1 12) are higher than 20°, preferably higher than 28°.

18. Lens (100) according to any of claims 4 to 17, characterized in that none of the draft facets (1 10) is parallel to a chief-ray (Rc).

19. Lens (100) according to any of claims 4 to 18, characterized in that the exit pupil (28) is a volume centered on the optical axis (1 12) of the lens (100) at a distance comprised between 2 and 10 cm from the first surface (102) and confined within a diameter comprised between 2 and 10 mm around the optical axis (1 12).

20. Lens (100) according to any of claims 4 to 19, characterized in that the exit pupil (28) is a volume centered on the optical axis (1 12) of the lens (100) at a distance comprised between 3 and 20 times the average thickness of the lens and confined within a diameter around the optical axis (1 12) that is comprised between 5% and 30% of the external diameter of the lens.

21 . Lens (100) according to any of claims 4 to 20, characterized in that the receiver is the pupil (22) of a human eye (20).

22. Lens (100) according to any of claims 4 to 21 , characterized in that the entrance pupil (30) is a two-dimensional entrance pupil (30) arranged perpendicularly to the optical axis (1 12).

23. Lens (100) according to any of claims 4 to 22, characterized in that the entrance pupil (30) is at a distance from the second surface (104) comprised between 5 and 50 mm.

24. Lens (100) according to any of claims 4 to 23, characterized in that the entrance pupil (30) has a dimension in the direction perpendicular to the optical axis (1 12) that is comprised between 50% and 150%, of the dimension of the lens (100) along the same direction.

25. Lens (100) according to any of claims 4 to 24, characterized in that both the slope facets (108) and the draft facets (1 10) are substantially conical surfaces.

26. Lens (100) according to any of claims 4 to 25, characterized in that the first surface (102) is a continuous and uniformly curved or flat surface.

27. Lens (100) according to any of claims 4 to 26, characterized in that the first surface (102) comprises a series of alternated slope and draft facets.

28. Lens (100) according to any of claims 4 to 27, characterized in that the substrate (106) is made of germanium and/or the first and second surfaces (102, 104) are coated with germanium.

29. Lens (100) according to any of claims 4 to 28, characterized in that the lens is substantially flat with a diameter of about 10 cm and with a total field of view of about 140°.

30. Lens (100) according to any of claims 4 to 29, characterized in that the lens has a focal length which varies between 30 mm and 75 mm.

31 . Head-mounted display comprising a display and a lens, characterized in that the lens (100) is according to any of the claims 4 to 30, wherein the entrance pupil (30) comprises the display (30) and the exit pupil (28) encompasses the possible positions of a human pupil (22) with respect to the lens (100).

32. Head-mounted display according to claim 31 , characterized in that the dimensions and relative positions of the lens (100) and the display (30), and the draft angles (Q) of the lens (100) are such that: rays coming from the display (30) through the lens (100) and through partial reflection of light on the draft facets (1 10) do not reach the eye and/or rays coming from the eye through the lens (100) and through total internal reflection of light on the draft facets (1 10) do not reach the display (30).

33. Head-mounted display according to claim 31 or 32, characterized in that for a draft facet (1 10) at a distance to the optical axis (1 12) greater than a predetermined threshold (S2, S3), the draft angle (Q) is equal or less than 30°; for a draft facet (1 10) at a distance to the optical axis (1 12) greater than a predetermined threshold (S2, S3), the depth of the draft facet is of about 300 pm; and/or for a draft facet (1 10) at a distance to the optical axis (1 12) greater than a predetermined threshold (S2, S3), the draft angle (Q) is comprised between 18 and 30°.

34. Head-mounted display according to any of claims 31 to 33, characterized in that it comprises two displays (30) with a diagonal size between 50 and 80 mm, and two respective lenses (100) according to one of claims 1 to 25, wherein the optical axes (1 12) of the lenses (100) intersect each other forming an angle of at least 30°.

35. Head-mounted display according to any of claims 31 to 34, characterized in that it comprises at least one optical element (50) arranged between the display (30) and the lens (100).

36. Use of the lens (100) according to anyone of claims 4 to 30 in an imagery device, wherein the receiver is an image sensor and the entrance pupil (3, 30) is an environment that is to be captured.