Lens with off-axis curvature center

Abstract

An off-axis curvature center lens is characterized by an x-y-z coordinate system of the convergence-reducing lens, the off-axis curvature lens comprising a distance-vision region with a non-negative distance-vision optical power, having a front distance-vision surface with a center of front distance-vision curvature, and a rear distance-vision surface with a center of rear distance-vision curvature; and a near-vision region with an optical power within 0.5D of the distance-vision optical power, having a front near-vision surface with a center of front near-vision curvature, and a rear near-vision surface with a center of rear near-vision curvature; wherein at least one of an x-coordinate of the center of front near-vision curvature is nasal relative to an x-coordinate of the center of front distance-vision curvature, and an x-coordinate of the center of rear near-vision curvature is temporal relative to an x-coordinate of the center of rear distance-vision curvature.

Claims

1. A convergence-reducing lens, wherein: a central normal of the convergence-reducing lens defines a z-axis, and a central region of the convergence-reducing lens defines a tangential, centered x-y plane, together defining an x-y-z coordinate system of the convergence-reducing lens, the convergence-reducing lens comprising a distance-vision region with a non-negative distance-vision optical power, having a front distance-vision surface with a center of front distance-vision curvature, and a rear distance-vision surface with a center of rear distance-vision curvature; and a near-vision region with an optical power within 0.5D of the distance-vision optical power, having a front near-vision surface with a center of front near-vision curvature, and a rear near-vision surface with a center of rear near-vision curvature; wherein at least one of an x-coordinate of the center of front near-vision curvature is nasal relative to an x-coordinate of the center of front distance-vision curvature, and an x-coordinate of the center of rear near-vision curvature is temporal relative to an x-coordinate of the center of rear distance-vision curvature; wherein the front distance-vision surface and the rear distance-vision surface at an x-distance from a center of the coordinate system define a distance-vision surface convergence angle; the front near-vision surface and the rear near-vision surface at the x-distance from the center of the coordinate system define a near-vision surface convergence angle; and the near-vision surface convergence angle is smaller than the distance-vision surface convergence angle.

2. The convergence-reducing lens of claim 1, wherein: the convergence-reducing lens is configured so that at least one of the x-coordinate of the center of front near-vision curvature is nasal relative to the z-axis of the coordinate system, and the x-coordinate of the center of rear near-vision curvature is temporal relative to the z-axis of the coordinate system.

3. The convergence-reducing lens of claim 1, wherein: the near-vision region has an area larger than 5 mm.sup.2.

4. The convergence-reducing lens of claim 1, wherein: the near-vision region has an area larger than 10 mm.sup.2.

5. The convergence-reducing lens of claim 1, wherein: the near-vision optical power matches the distance-vision optical power within 0.25 D.

6. The convergence-reducing lens of claim 1, wherein: the distance-vision optical power and the near-vision optical power are less than 0.5 D.

7. The convergence-reducing lens of claim 6, wherein: the distance-vision optical power and the near-vision optical power are 0 D.

8. The convergence-reducing lens of claim 1, wherein: a majority of the near-vision region is located in a lower-inferior nasal quadrant of the convergence-reducing lens.

9. The convergence-reducing lens of claim 1, comprising: a front surface, having a distance-vision front-tangential at an x-distance from the center of the coordinate system, and a near-vision front-tangential at the same x-distance; and a rear surface, having a distance-vision rear-tangential at the same x-distance, and a near-vision rear-tangential at the same x-distance; wherein the distance-vision front-tangential and the distance-vision rear-tangential make a distance-vision surface convergence angle, and the near-vision front-tangential and the near-vision rear-tangential make a near-vision surface convergence angle, wherein the near-vision surface convergence angle is smaller than the distance-vision surface convergence angle.

10. The convergence-reducing lens of claim 1, wherein: the distance-vision region has a distance-vision z-axis; the near-vision region has a near-vision z-axis, wherein the near-vision z-axis is rotated in a nasal direction relative to the distance-vision z-axis.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIGS. 1A-B illustrate the refraction angle of optical lenses.

(2) FIGS. 2A-C illustrate the refraction angles of a monovision lens.

(3) FIGS. 3A-B illustrate the induced refraction, increasing the gaze convergence angle.

(4) FIGS. 4A-B illustrate an effect of a convergence-reducing lens.

(5) FIGS. 5A-D illustrate light propagation in convergence-reducing lenses.

(6) FIGS. 6A-B illustrate embodiments of a convergence-reducing lens.

(7) FIGS. 7A-D illustrate embodiments of a convergence-reducing lens.

(8) FIGS. 8A-C illustrate contour-representations of various convergence-reducing lenses for near-zero optical powers.

(9) FIGS. 9A-D illustrate contour-representations of various convergence-reducing lenses for optical power D.

(10) FIGS. 10A-C illustrate various designs of the near-vision region.

(11) FIGS. 11A-B illustrate lens designs.

(12) FIGS. 12A-B illustrate off-axis centers of curvatures in various embodiments of the convergence-reducing lens.

DETAILED DESCRIPTION

(13) Embodiments of the invention are placed into context by first describing how regular, positive power lenses of existing spectacles induce increased gaze-convergence angles for near-vision, thus exacerbating already existing digital eyestrain. This will be followed by the description of the embodiments of the invention.

(14) FIG. 1A illustrates how a typical, positive power mono-vision optical lens 1 effects incident light rays 2. When parallel rays 2 are incident on the lens 1, the lens 1 focuses them into a focus point F.

(15) FIG. 1B zooms in on an off-center, or off-axis portion of the incident light rays. Visibly, the off-center, or off-axis parallel rays are refracted towards the focus point F by the angled front surface and the angled back surface of the lens 1 according to well-established laws of light-refraction. The overall effect of the light rays propagating through these two angled surfaces is that they get refracted by an induced angle of refraction .

(16) There are different, related ways to characterize the amount of refraction by a lens region at a radial distance r from the axis. One characterization is by the refraction angle itself Another is by the tangent of this same refraction angle, expressed as a ratio of r, the radial distance of the region of the lens from a lens optical axis 3, to f, the focal distance of the lens:
tan =r/f.(1)

(17) This relation shows that a lens of optical power of D diopters, defined as D=1/f[1/m], induces a refraction angle for rays that are incident at the lens at a radial distance r from the axis 3 of the lens 1, where is given by
tan =r*D.(2)

(18) FIG. 2A illustrates a simple mono-vision lens 10 with optical power D.

(19) FIG. 2B illustrates the above notion that the angled surfaces of the mono-vision lens 10 of FIG. 2A induce a refraction angle in regions away from the axis of the lens. Since the magnitude of the refraction angle depends only on the radial distance from the axis, the iso- contours, i.e. the points where the magnitudes of the refraction angle are equal, form concentric circles. The shown circles have radii of about r=0.8 mm, r=1.6 mm, 2.4 mm, r=3.2 mm, and r=4.0 mm. Equation (2) provides that tan , the tangent of the refraction angle is given as the product of the radius r and the optical power D. Here, the units of D are [1/m], and the units of r are [m]. Since typical values of r are 1-20 millimeters, and values of D are a couple diopters [1/m], tan is typically of the order of 10.sup.3-10.sup.2 radian, which translates to a few-to-a-few-tens of arc-minutes. As an example, for r=1 mm, and D=1[1/m], tan =1*10.sup.3 radian=3.5 arc-minutes. At small angles, tan is well approximated by . Therefore, returning to FIG. 2A, on the shown circles the deflection angle takes the values 2.8D, 5.6D, 8.4D, 11.2D and 14D, in arc minutes.

(20) Finally the lower graph of FIG. 2B illustrates .sup.x, the x-component of the refraction angle , where the x-y coordinate system is based at the center of the lens 10, with its axes being horizontal and vertical in the plane of the lens 10, as shown. There are several ways to define .sup.x that are analogous to each other. One practical definition is .sup.x=sin *, where is the angle measured from the negative, or lower half of the y-axis, pointing straight down in the plane of FIG. 2B, as shown. Combining with Eq. (2), and using that sin * r=x, the x coordinate of the radial vector of length r, one gets the simple relation:
.sup.x=x*D.(3)

(21) The lower graph in FIG. 2B shows .sup.x as a function of the angle in general. The upper figure shows the particular values of .sup.x along the +45 degree and along the 45 degree lines, using sin(45)=0.7. These values are: .sup.x=2D, 4D, 6D, 8D and 10D, as shown.

(22) Analogous definitions of .sup.x include tan .sup.x=sin * tan , which accounts more precisely for the geometry of projections of a refracted light 2. However, for the present small angles these two definitions yield very similar mathematical relations and numerical values. Finally, the formula can be extended for light rays 2 that are not parallel with the optical axis 3, but, rather, make an angle with the optical axis 3. In general, such extensions would yield an object-angle dependent expression, through a dependence on the angle . Notably though, such a -dependent formula can be expanded in . Such an expansion would reproduce Eq. (2) in leading order in .

(23) .sup.x characterizes the s-component of the refraction angle that in turn determines how much a wearer of the spectacle need to turn her/his gaze to concentrate on these light rays. The larger the .sup.x in a region of the lens, the more the light 2 passing through this region is refracted by the lens and the more a wearer has to turn her/his gaze.

(24) FIG. 2C illustrates the iso-.sup.x contours for the lens 10 of FIG. 2B, where .sup.x assumes the same values. Visibly, for the mono-vision lens 10 of optical power D, the iso-.sup.x contours are approximately straight lines parallel to the y-axis, since .sup.x only depends on the x coordinate of the points of the contour. For larger optical powers and radii, where the linear approximations start to garner corrections, the iso-.sup.x contours start bulging radially outward close to the x-axis,

(25) FIGS. 3A-B illustrate how the refraction angle, induced by a positive power lens, impacts the convergence of the gaze of glass-wearers, based on these general considerations.

(26) FIG. 3A illustrates that when a person is gazing at a remote object, then the gazing-directions of the left and right eves are essentially parallel, and thus there is no convergence of the gazing-directions and the eye muscles are not strained at all. In such cases, the two z-axes 3, pointing to the remote object through the center of the eye 5, coincide with the two eye-optical axes 9. The light 2 from the remote object is entering the eye 5 through the eye-lens 7 on its way to hit the retina of the eye 5. These parallel axes will be used as references to characterize the gaze-convergence angles of gazes directed at near objects through various lenses next.

(27) FIG. 3B illustrates that when a person is gazing at a near object, the gazes of the left and right eyes are tilted, or rotated, towards each other, each gaze making a non-zero gaze-convergence angle with the z-axes 3. Since the gaze-convergence angle characterizes the convergence of the gazes of the two eyes towards each other, in what follows the gaze-convergence angle will refer specifically to the x-component of the overall gaze-rotation angle of the eye. This makes the gaze-convergence angle analogous to the x-component of the refraction angle .sup.x, while also simplifying the terminology.

(28) As mentioned before, the eyeballs are rotated by muscles attached externally to the eye. In particular, the lateral, x-directional, rotations are controlled by the medial rectus and the lateral rectus muscles, and the vertical rotations are controlled by the superior rectus and the inferior rectus, and inferior oblique muscles. When the medial rectus muscles of the left-eye and the right-eye contract, the gazes of these eyes converge towards each other. A person, training his/her eye on a near object, such as an electronic screen, a digital screen, a screen of a mobile electronic device, work-related papers, or even a book, for extended periods requires the continuous contraction of the medial rectus muscles, thus exerting substantial strain on them. This digital eyestrain can lead to fatigue, leading to headache, eventually culminating in migraines, caused by the demands of the modern, digital lifestyle.

(29) The digital lifestyle can induce other forms of asthenopia, or eye-strain, and other types of convergence-disorders, including proprioceptive disparity, and fixation disparity. Proprioceptive disparity is an imbalance between where the eyes are consciously focused and the nonvisual perception of where the object is located in space. This disparity often varies with space. The brain of a patient with a proprioceptive disparity can compensate this disparity to a degree in order to maintain a clear image of the target. However, when the disparity becomes too big to be compensated, the trigeminal nerve can get overstimulated, resulting in patients experiencing headaches, eye fatigue, pain around the eyes, blurred vision, neck pain, dry eyes, and other general symptoms of asthenopia.

(30) A class of symptoms especially worthy of mentioning is Computer Vision Syndrome (CVS), which is estimated to affect more than 100 million Americans. Computer Vision Syndrome is the physical eye discomfort felt after a prolonged amount of time in front of digital devices at near, causing an array of unnecessary symptoms, and can have effects on productivity.

(31) Another large class of symptoms is known by the name of Chronic Daily Headaches (CDH). CDH symptoms are estimated to affect more than 30 million Americans. These patients suffer from an over-stimulation of the trigeminal nerve that manifests itself in the form of chronic daily headaches. Various factors and triggers are believed to contribute to the debilitating issue of chronic daily headache. As a result, patients suffering from CDH are limited to treatment options that merely seek to dull the symptoms. A large subset of chronic daily headache patients (believed to be as large as 33% of the population) shows objective signs of a misalignment between how the central visual system, peripheral visual system and neurological system interact.

(32) FIG. 4A illustrates that positive power spectacles 10 can exacerbate the symptoms of eye-strain, astenophia, Computer Vision Syndrome and proprioceptive disparity, caused by digital devices, because looking at digital devices, or objects 11 that are nearby forces the wearer to gaze through the lower-inferior nasal quadrant, the near-vision region, of their glasses. As shown before, in this off-center near-vision region positive power lenses 10 refract the light by a refraction angle , as described by Eqs. (1)-(3). A ray that propagates front the near object 11 to the retina with the refraction angle a forces on the wearer a greater gaze-convergence angle than a ray that propagates from the same object to the same retina but without the refraction angle . Therefore, positive power lenses 10 force an increased gaze-convergence angle and thus cause an increased strain on the medial rectus muscles when the wearer is looking at near objects. The sustained and excessive contraction of the medial rectus muscles increases the tendencies for a digital migraine that affect and possibly debilitate the wearer.

(33) FIG. 4B illustrates embodiments of a convergence-reducing lens 100 in a convergence-reducing spectacle 100 that can reduce, and often eliminate the symptoms caused by eye-strain, astenophia, Computer Vision Syndrome and proprioceptive disparity. The convergence-reducing spectacles 100 with convergence-reducing lenses 100 have a suitably modified refraction-angle that reduces the gaze-convergence angle when their wearers look at a nearby object, such as at a digital device. Reduced gaze-convergence angles require a lesser rotation of the eyes in the nasal direction, and therefore relieve the continuous contraction and strain of the medial rectus muscles of the eyes. This reduced muscle strain reduces and often eliminates digital migraines.

(34) FIGS. 5A-B illustrate in detail an embodiment of an eye-strain-reducing lens 100, or convergence-reducing lens 100 that relieves eye-strain and related digital eye discomforts. Throughout this application, the terms eye-strain-reducing lens and convergence-reducing lens will be used and treated interchangeably. For clarity, only one of the convergence-reducing lenses 100 of the convergence-reducing spectacle 100 are shown. The below description applies for the other lens of the convergence-reducing spectacle 100 with the appropriate modifications. A nose of the spectacle-wearer is shown for reference. Embodiments of the convergence-reducing lens 100 can define an x-y-z coordinate system as follows. A central normal of the convergence-reducing lens 100 can define a z-axis 3, and a central region of the convergence-reducing lens 100 can define a tangential, centered x-y plane. The center of the coordinate system can be at the center of the lens 100. The convention is adopted that the x-axis is horizontal with respect to the spectacle 100, and thus goes through the centers of both the left and the right convergence-reducing lenses 100. Accordingly, the y-axis is vertical.

(35) With this coordinate system, and with further reference to FIG. 8A, the convergence-reducing lens 100 can include a distance-vision region 110, having a non-negative distance-vision optical power, configured to refract a light ray 2, directed by a source 11 at a distance-vision region point P.sub.d at a distance-vision x-distance x.sub.Pd from a center of the coordinate system, to propagate to an eye-center representative location 8. In some embodiments, the eye-center representative location 8 can be an eye center 8 itself. In other embodiments, it can be a sensor, positioned at the location 8, or a screen, positioned across location 8, wherein the eye-center representative location 8 lies on the z-axis 3, at a z-distance in the range of 15-25 mm from the center of the lens's coordinate system in a direction that is opposite to the direction of the source. These latter eye-center representative locations 8 can be more suitable and accessible for measurements and characterizations.

(36) The convergence-reducing lens 100 can further include a near-vision region 120, having a near-vision optical power that matches the distance-vision optical power within 0.5 diopters D, configured to refract a light ray 2, directed by the source 11 at a near-vision region point P.sub.n at a near-vision x-distance x.sub.Pn from the center of the coordinate system, to propagate to the eye-center representative location 8. Since the optical power of the near-vision region 120 can be very close, and in some embodiments, equal to the optical power of the distance-vision region 110, embodiments of the convenience-reducing lens 100 can be called a mono-vision lens, or a single-vision lens. This aspect can distinguish these lenses from other, traditional bi-focal lenses where the near-vision and distance-vision optical powers are different.

(37) For clarity, in this document the term optical power refers to the optical power specifically related to the focal length f of the lens, and is measured in diopters D that are inversely related to the focal length: D=l/f. Also, FIG. 5A can be a cross section of the convergence-reducing lens 100 at a high, positive y coordinate, whereas FIG. 5B may illustrate a cross section of the same convergence reducing lens 100 at a lower, negative y coordinate.

(38) In embodiments, the near-vision x-distance x.sub.Pn is smaller than the distance-vision x-distance x.sub.Pd, as shown. Visibly, since in these embodiments the near-vision x-distance x.sub.Pn is smaller than the distance-vision x-distance x.sub.Pd, the wearer of this convergence-reducing lens 100 can rotate an eye-optical axis 9 of his/her eye closer toward the z-axis 3 when looking at the source 11 through the near-vision region 120 relative to the case when the wearer is looking at the same source 11 through the distance-vision region 110, thereby reducing the gaze convergence angle, as described further next. As indicated in FIG. 5B, the reduced gaze-convergence angle translates into a stress-reducing rotation of the eye 5. Accordingly, the convergence-reducing lens 100 can also be called an eye-strain reducing lens 100. For this reason, the convergence-reducing spectacles 100 deliver the much-needed medical benefit of reducing eye-strain, digital migraines, proprioceptive disparity, fixation disparity, asthenopia, and convergence-disorders in general.

(39) A first inventive layer of the described technologies involves bifocal glasses, which already have a near-vision region separate from the usual distance-vision region. Such glasses can be bestowed with the additional medical benefit of eye-strain reduction by making the convergence properties of these two vision regions also different.

(40) Beyond this layer, a distinguishing feature of the here-described single-vision, or monovision convergence-reducing lenses 100 is that they have a near-vision region 120 with a refractive power different from the refractive power of the distance-vision region 110, in spite of the two regions having matching optical powers. This is to be contrasted with bifocal lenses, where both the. refractive and the optical powers of the two vision regions are different. This is a qualitative, crucial distinction for at least the following reasons.

(41) (1) Bifocal spectacles already have two vision regions with a differing optical property, the optical power. Therefore, it may occur to a lens designer to make a further optical property also different, such as the refractive power, to reduce convergence. However, in monovision lenses it is far from obvious for a designer to think of and to create a near-vision region for the sole purpose of delivering a different refractive power, while making sure that the near-vision region retains the same optical power as the rest of the lens.

(42) (2) The global market for spectacle lenses exceeded 1 billion units sold worldwide in 2015, and more than 320 million in the US alone. It is also estimated that 75% of the US population, or about 240 million people wear some sort of vision correcting spectacles. By far the broadest market segment of spectacles sold in the US today, about 90% of the total market, have single vision lenses, and only about 10%, or 20-25 million people wear bifocals. The mostly younger and early-middle age wearers of single-vision tenses simply do not need bifocal lenses. Some industry surveys estimate the number of people who suffer, or report, Computer Vision Syndrome to exceed 100 million people. Therefore, introducing convergence-reducing near-vision regions into single vision spectacles will extend the reach of the convergence-reduction technology from the narrow, 10-20 million unit/year market segment of bifocals to the 100 million-plus unit/year market segment of monovision glasses. Therefore, the here-described monovision glasses will dramatically broaden the group of people to whom the medical benefit of convergence-reduction can be delivered.

(43) (3) Convergence-reducing monovision glasses with zero or near zero optical powers will qualitatively broaden the market penetration to yet another wide class. These glasses will deliver the medical benefit of convergence reduction to people who do not need optical power correction and therefore did not think of wearing glasses up to now. For this reason, zero optical power monovision spectacles will dramatically extend the segment of the population to whom the medical benefit of convergence-reduction is delivered even further.

(44) Finally, it is mentioned that in present-day optometric practice, most doctors have a different theory of the cause of eye-strain, and therefore offer very different treatments and procedures to alleviate eye-strain, or asthenopia. Optometrists often prescribe switching to glasses with blue light filters, or suggest using humidifiers. Therefore, prescribing glasses with the here-described convergence-reduction technology rests on a very different medical insight regarding what causes eye-strain, and an inventive treatment to alleviate it that is genuinely different from what is prescribed by today's optometric practitioners.

(45) Here and later in the text, the light propagation is described as originating by the source 11, or from an object 11, interchangeably. The source 11 can be a laser pointer or other directed light source that actively generates a light ray 2. In some other embodiments, the object 11 may not be an active light source, rather, an object or mirror that reflects a light in the described direction, wherein the light originated somewhere else. From the viewpoint of the light propagation, these two cases can be interchangeable. The object 11, or source 11, can be at a z-distance z.sub.o/s from the x-y plane of the convergence-reducing lens 100.

(46) In embodiments of the convergence-reducing lens 100, the distance-vision region 110 can be configured to refract the light ray 2, directed by the source 11, or object, 11 at the distance-vision region point P.sub.d at the distance-vision x-distance x.sub.Pd, to intersect a y-z plane of the coordinate system with a distance-vision gaze-convergence angle .sub.d; whereas the near-vision region 120 can be configured to refract the light ray 2, directed by the source 11 at the near-vision region point P.sub.n at the near-vision x-distance x.sub.Pn, to intersect the y-z plane with a near-vision gaze-convergence angle .sub.n. In these embodiments of the convergence-reducing lens 100 the near-vision gaze-convergence angle .sub.n can be smaller than the distance-vision gaze-convergence angle .sub.d. Typically, the intersection of the refracted light 2 with the y-z plane with the gaze convergence angle .sub.n/d occurs at the eye-center representative location 8.

(47) Here, the gaze-convergence angles .sub.d and .sub.n characterize the convergence of the left and right eye's gaze, and thus they can correspond to the x-component of the overall, 3d dimensional rotation angle of the eyes, in analogy to .sup.x, the x-component of the overall refraction angle .

(48) This is a second expression that when the wearer looks at an object 11 through the near-vision region 120 of the convergence-reducing lens 100, she/he does not need to rotate her/his eyes away from the z-axis 3 as much as in the case of looking at the same object through the distance-vision region 110 of the lens 100. Therefore, embodiments of the convergence-reducing lens 100 indeed reduce the convergence angle of the gaze of its wearer, when looking at objects through the near-vision region 120, compared to looking through the distance-vision region 110, or even through an analogous regular positive power lens 10.

(49) In some embodiments of the convergence-reducing lens 100, the distance-vision region 110 can be configured to refract the light ray 2, directed by or from the source 11 at the distance-vision region point P.sub.d at the distance-vision x-distance x.sub.Pd, by a distance-vision refraction angle .sub.d, whereas the near-vision region 120 can be configured to refract the light ray 2, directed by or from the source 11 at the near-vision region point at the near-vision x-distance x.sub.Pn, by a near-vision refraction angle .sub.n. In such embodiments of the convergence-reducing lens 100, an x-component .sub.n.sup.x of the near-vision refraction angle .sub.n can be smaller than an x-component .sub.d.sup.x of the distance-vision refraction angle .sub.d. This is a third expression that the lens 100 is reducing the gaze-convergence when its wearer is looking at the object 11 through the near-vision region 120, relative to looking at the same object 11 through the distance-vision region 110.

(50) The above three expressions of the gaze-convergence reducing aspects of the convergence-reducing lens 100 are stated as boxed inequalities in FIG. 5B. These inequalities are repeated here:
x.sub.Pn<x.sub.Pd,(4)
.sub.n<.sub.d, and(5)
.sub.n.sup.x<.sub.d.sup.x.(6)

(51) Embodiments of the convergence-reducing lens 100 satisfy at least one of these three inequalities (4)-(6).

(52) The above descriptions of embodiments of the convergence-reducing lens 100 also articulate auditing protocols to determine whether a lens is a convergence-reducing lens. (1) It is possible to measure the described distances x.sub.Pd and angles .sub.d.sup.x and .sub.d directly when a wearer of the lens is looking at an object through a potential distance-vision region of a lens, followed by measuring the corresponding distances and angles .sub.n.sup.x and .sub.n as the wearer looks thiough a potential near-vision region of the lens, and then to compare the measured angles and distances to verify whether they satisfy at least one of the described three inequalities. For potential lenses, where the changes of the angles are small, an eye-tracking or eye-imaging system can be used to determine the changes in the wearer's gaze-angle to detect the small changes and differences. (2) Instead of measuring angles and directions of a wearer's gaze, an eye model with realistic parameters can be used as well. The eye model can include a disk of a diameter of about 20-25 mm, such as 24 mm, rotatable around a y-axis at an eye-center representative location 8. The front of the eye model can be, positioned 10-15 mm behind the lens 100, the eye-center representative location 8 about 20-30 mm behind the lens 100. The eye model can include an appropriate eye lens 7, with a total optical power approximately equal to that of the cornea, about 40-45 D, plus that of the lens, about 15-25 D. A directed light source, such as a laser pointer or equivalents can be deployed in place of the source 11 and its light can be pointed at the potential distance-vision region and near-vision region of an audited lens so that after refraction by the lens the light goes through the eye-center representative location 8 of the eye model in both cases. The described angles and distances can then be measured to determine whether at least one of the three above inequalities applies.

(53) (3) Finally, measurements without involving a wearer's eye, or even an eye-model, can also be sufficient to determine whether a lens is an embodiment of the convergence-reducing lens 100. A lens can be audited on a fixed optical table by pointing a laser pointer from a position of the source 11 at the lens such that its light after refraction by the lens propagates through a candidate point for an eye-center representative location 8, about 20-30 mm behind the center of the lens 100 along the z-axis 3. The light's propagation can be tracked, e.g., by implementing a screen in the y-z plane of the lens 100 on the side opposite to the source 11. The light of the laser pointer 11 can be directed at a potential distance-vision region of the audited lens and through a potential near-vision region of the audited lens, ensuring that the refracted light in both cases intersects the y-z plane at the same z-distance from a center of the coordinate system that is representative of an eye center 8. As described above, such representative locations can be 20-30 mm behind the center of the lens, on the z-axis 3. Once the angles and distances, discussed before, are measured for the light directed at the potential distance-vision and then the potential near-vision regions, a lens is an embodiment of the convergence-reducing lens if at least one of the three inequalities in FIG. 5B, and discussed above in Eqs. (4)-(6), holds for the measured angles and distances. Other auditing protocols will be described later, in relation to FIGS. 5C-D and FIGS. 7A-D.

(54) FIGS. 5A-B illustrate that the object/source 11 can be a near object, located at a source x-distance from the z-axis 3 of the coordinate system that is larger than a radius of the convergence-reducing lens 100, and at a source z-distance that is between 10 cm and 100 cm. Such an off-center, off-axis source 11 can be a good representation of a near object, aligned with a nose of the spectacle wearer, as shown.

(55) FIGS. 6A-B illustrate that in other embodiments, the object 11 can be farther away. For example, the source/object 11 can be located at a source x-distance from the z-axis 3 of the coordinate system that is smaller than a radius of the convergence-reducing lens 100; and at a source z-distance that is larger than 100 cm. A class of these objects/sources 11. can include light from a laser pointer, directed in parallel to the z-axis 3 at the near-vision region point P.sub.n and the distance-vision region point P.sub.d. Embodiments of the convergence-reducing lens 100 satisfy at least one of three inequalities that are the analogs of the three inequalities of FIGS. 5A-B, shown in the three boxes of FIG. 6B. The locations of the sources 11 are somewhat different in the embodiments of FIGS. 5A-B and FIGS. 6A-B, therefore the distance and angle ranges where the two sets of inequalities are satisfied may not be exactly equal. Nevertheless, the validity ranges of the inequalities largely overlap, and therefore, both sets of inequalities are representations of embodiments of the convergence-reducing lens 100.

(56) FIGS. 5C-D illustrate further aspects of the convergence-reducing lenses 100. The characterization of the embodiments of FIGS. 5C-D is largely analogous to that of FIGS. 5A-B, since the characterization is motivated by the reversibility of the paths of propagating light rays 2. To indicate that elements in FIGS. 5C-D are related to the elements in FIG. 5A-B by path-reversal, corresponding labels are used, with an r added, where appropriate. With these introductory considerations, some embodiments of a convergence-reducing lens 100 can have a distance-vision region 110, having a non-negative distance-vision optical power, that is configured to refract a light ray 2 directed by a source 8r at a distance-vision region point P.sub.d, at a distance-vision x-distance X.sub.Pd from the center of the coordinate system, to propagate to an image point 11r, or object/source 11r. The image point 11r. in some sense the reverse-pair of the object/source 11 of the embodiments in FIGS. 5A-B, can be located at a z-distance z.sub.l from the x-y plane of the lens 100. The source 8r, in some sense the reverse-pair of the eye-center representative location 8 of the embodiments in FIGS. 5A-B, can be located on the z-axis 3 at a source-z-distance z.sub.s from a center of the coordinate system.

(57) This embodiment of the convergence-reducing lens 100 can further include a near-vision region 120, having a near-vision optical power that matches the distance-vision optical power within 0.5D, configured to refract a light ray 2 directed by the source 8r, located at the same source-z-distance z.sub.s from a center of the coordinate system, at a near-vision region point P.sub.n at a near-vision x-distance x.sub.Pn from the center of the coordinate system to propagate to the same image point 11r. In these embodiments, the near-vision x-distance x.sub.Pn can be smaller than the distance-vision x-distance x.sub.Pd, in analogy to inequality (4) of the embodiments of FIGS. 5A-B.

(58) In some embodiments, the distance-vision region 110 can be configured so that the source 8r can direct the light ray 2 to propagate to the image point 11r via a refraction at the distance-vision region point P.sub.d by directing the light ray 2 with a distance-vision gaze-convergence angle .sub.d relative to a y-z plane of the coordinate system; and the near-vision region 120 can be configured so that the source 8r can direct the light ray 2 to propagate to the same image point 11r via a refraction at the near-vision region point P.sub.n by directing the light ray with a near-vision gaze-convergence angle .sub.n relative to the y-z plane of the coordinate system. In these embodiments, the near-vision gaze-convergence angle .sub.n can be smaller than the distance-vision gaze-convergence angle .sub.d, in analogy to inequality (5) above.

(59) In some embodiments, the distance-vision region 110 can be configured to refract the light ray 2, directed by the source 8r at the distance-vision region point P.sub.d to propagate to the image point 11r, by a distance-vision refraction angle .sub.d. The near-vision region 120 can be configured to refract the light ray 2, directed by the source 8r at the near-vision region point P.sub.n to propagate to the same image point 11r, by a near-vision refraction angle .sub.n. In embodiments, .sub.n.sup.x, the x-component of the near-vision refraction angle .sub.n can be smaller than .sub.d.sup.x, the x-component of the distance-vision refraction angle, .sub.d, in analogy to inequality (6) above.

(60) FIGS. 8A-E illustrate a frontal view of an embodiment of the convergence-reducing lens 100, looking onto the x-y plane of the lens from the z-axis direction. FIG. 8A shows iso-dioptric contour lines of the optical power, whereas FIG. 8B shows iso-.sup.x refraction angle contour lines of the convergence-reducing lens 100. In some embodiments of the convergence-reducing lens 100, the distance-vision region 110 can have a distance-vision optical power of D diopters. The near-vision region 120 can have a near-vision optical power that matches the distance-vision optical power within 0.5 D; and a channel region 115 that can connect the distance-vision region 110 and the near-vision region 120. In some embodiments, the near-vision optical power can match the distance-vision optical power within 0.25 D. Because of the close matching of the distance-vision optical power and the near-vision optical power, such embodiments can be called mono-vision lenses, single-vision lenses, or mono-focal lenses. The channel region 115 can have an optical power that matches the distance-vision optical power and the near-vision optical power within 0.5 D. In designs, where the distance-vision optical power and the near-vision optical power are the same, the channel region optical power can also have this shared value. In designs, where the distance-vision optical power and the near-vision optical power differ by a small amount, such as by less than 0.5 D, the channel region optical power can smoothly interpolate between these near-equal optical powers.

(61) As shown in FIG. 8A, in some embodiments, the distance-vision optical power and the near-vision optical power can be near zero, such as less than 0.5 D. In some embodiments, the distance-vision and the near-vision optical power can be zero diopter, 0 D.

(62) Such 0 D convergence-reducing lenses 100 can be used by persons who do not need a correction of the optical power of their eyes, yet still feel a digitally caused strain on their eyes, a digital eyestrain, that is caused by extended periods of gazing at near objects, such as digital, electronic, or computer screens. Such persons may decide to wear convergence-reducing spectacles 100 that reduce their digital eyestrain even if they do not need an optical power correction.

(63) Embodiments of the convergence-reducing lens 100 can further include a nasal transition region 135n and a temporal transition region 135t. In these regions, the optical power may deviate from 0 D for reasons that are explained below.

(64) In some embodiments, an area of the near-vision region 120 can be larger than 5 mm.sup.2. In some embodiments, the area of the near-vision region 120 can be larger than 10 mm.sup.2.

(65) FIG. 8B illustrates that in some embodiments of the convergence-reducing lens 100 with near 0 D optical power, .sub.d.sup.x, an x-component of the distance-vision refraction angle .sub.d in the distance-vision region 110 can be near 0 as well, because .sub.d itself is near zero, based on Eqs. (2)-(3) and recalling that the optical power is 0 D. In these embodiments, .sub.n.sup.x, an x-component of the near-vision refraction angle .sub.n in the near-vision region 120 can be positive. The magnitude of the refraction angles was discussed previously. In many embodiments .sup.x can fall in the range of 0.5-50 arc minutes, in some embodiments in the range of 1-10 arc minutes. In FIG. 8B, .sub.n.sup.x=+6 arc minutes in the near-vision region 120 as indicated by underlining, the value.

(66) These values are to be taken at the same x-distances from the center of the coordinate system for the distance-vision region 110 and for the near-vision region 120. This is shown by the near-vision region point P.sub.n being a reflection of the distance-vision region point P.sub.d across the x-axis, and thus having the same x-distance from the center of the coordinate system.

(67) Finally, these lenses may include a progression region 130, at least partially between the distance-vision region 110 and the near-vision region 120, wherein a light ray 2, directed from the source 11 at a progression region point at a progression x-distance is refracted to propagate to the eye-center representative location 8, wherein the progression x-distance is between the near-vision x-distance x.sub.Pn and the distance-vision x-distance x.sub.Pd. Such progression regions 130 are also characterized by .sub.p.sup.x, an x-component of a progression refraction angle .sub.p that progresses between the x-components of the distance vision refraction angle .sub.d.sup.x and the near-vision refraction angle .sub.n.sup.x. In the shown example, .sub.p.sup.x progresses between .sub.d.sup.x=0 and .sub.n.sup.x arc minutes. It is noted that, at least in some embodiments, the progression region 130 need not coincide with the channel region 115 of FIG. 8A.

(68) In FIG. 8A, the transition regions 135n and 135t can emerge for the following reason. In general, transition regions like the nasal transition region 135n and temporal transition region 135t are formed between the distance-vision region 110 and the near-vision region 120 when their optical properties are different in some aspect. This optical property can be their optical power, cylinder, or astigmatism. This difference in optical properties can lead to undesirable optical distortions. The transition regions 135n/t are designed to minimize these distortions. In the presently described convergence-reducing lenses 100, the optical power of the distance-vision region 110 and the near-vision region 120 can be close, or even the same.

(69) These convergence-reducing lenses 100, however, have different refraction angles .sub.d and .sub.n in the corresponding distance-vision region 110 and near-vision region 120. This difference may induce optical distortions. For this reason, it may reduce the optical distortions in these lenses 100, driven by the difference of the refraction angles .sub.d and .sub.n, to include the transition regions 135n/t, and the progression region 130 to smoothly interpolate between the .sub.d and the .sub.n refraction regions. FIG. 9A shows that in some embodiments, only a nasal transition region 135n may suffice for this purpose.

(70) FIG. 8B shows that in some embodiments, the majority of the near-vision region 120 can occupy the lower, or inferior nasal quadrant of the lens 100. In some embodiments, the near-vision region 120 can extend to the lower temporal quadrant, as shown.

(71) FIG. 8C shows that in some embodiments, the near-vision region 120 may fill even the lower nasal quadrant only partially.

(72) FIG. 9A shows a convergence-reducing lens 100 that has an optical power D. FIGS. 9B-D show the iso-.sup.x contours for various embodiments of the convergence-reducing lens 100 of FIG. 9A. As discussed in relation to FIG. 2C, in lenses with a fixed optical power, the iso-.sup.x contours may be vertical lines.

(73) FIG. 9B illustrates an embodiment of lens 100 where the near-vision region 120 fills out the lower nasal quadrant only partially. FIG. 9C illustrates an embodiment of lens 100 where the near-vision region 120 fills out the lower nasal quadrant fully. FIG. 9D illustrates an embodiment of the lens 100 where the near-vision region 120 fills out the lower nasal quadrant and also extends into the lower temporal quadrant.

(74) FIGS. 9B-D also illustrate that embodiments of the convergence-reducing lens 100 can compensate and reduce the refraction by the positive optical power distance-vision region 110 so well in the near-vision region 120 that the negative x-component of the distance-vision refraction angle .sub.d.sup.x can be compensated into a smaller-magnitude negative .sub.n.sup.x in FIG. 9B, a zero .sub.n.sup.x in FIG. 9C, or even into an overcompensated, opposite sign, positive .sub.n.sup.x in FIG. 9D. Such an overcompensated case was already illustrated with the light rays of FIG. 4B, with a refraction angle of the opposite sign than in FIG. 4A.

(75) In the special case when the optical power of the distance-vision region 110 is approximately zero, .sub.d.sup.x is accordingly small or zero. In such cases, the near-vision region can compensate a near-zero x-component of the distance-vision refraction angle .sub.d.sup.x into a positive .sub.n.sup.x.

(76) FIGS. 10A-C illustrate that the near-vision region 120 can have different shapes, including an oval, a quadrant, a triangle, a rectangle, an elongated region, a diagonal region, a channel or a corridor.

(77) FIGS. 11A-B illustrate two, embodiments of the convergence-reducing lens 100 that can achieve and deliver the above described properties of the convergence-reducing lens 100, in particular, that show configurations and designs of lenses 100 that satisfy at least one of the earlier described three inequalities (4)-(6).

(78) FIG. 11A illustrates that embodiments of the convergence-reducing lens 100 can include a front surface 140f, with a distance-vision front-tangential 145fd at an x-distance from the center of the coordinate system, and a near-vision front-tangential 145fn at the same x-distance; and further they can include a rear surface 140r, with a distance-vision rear-tangential 145rd at the same x-distance, and a near-vision rear-tangential 145rn at the same x-distance. These tangentials 145 are indicated by dashed lines. The distance-vision front-tangential 145fd and the distance-vision rear-tangential 145rd form a distance-vision surface convergence angle .sub.dvr, while the near-vision front-tangential 145fn and the near-vision rear-tangential 145rn make a near-vision surface convergence angle .sub.nvr. In FIG. 11A, the front and rear surfaces 140f and 140r of the near-vision region 120 are the indented surfaces close to the center of the lens 100, and therefore the near-vision surface convergence angle .sub.nvr is smaller than the distance-vision surface convergence angle .sub.dvr:
.sub.nvr<.sub.dvr.(7)

(79) This inequality is one way to design a convergence-reducing lens 100 that achieves at least one of the three inequalities (4)-(6). Several designs can be consistent with this inequality. In some cases, the inequality of the angles in Eq. (7) can be solely driven by one of the tangentials being different, and the tangential of the other surface being the same for the front and rear, surfaces. In some cases, the lens 100 can be a meniscus lens 100. It is also noted that these angles .sub.nvr and .sub.dvr depend on the x-distance where the tangentials were fitted to the surfaces 140r and 140f: .sub.nvr=.sub.nvr(x) and .sub.dvr=.sub.dvr(x). The angles .sub.nvr(x) and .sub.dvr(x) are to be determined and compared at the same x-distances from the center.

(80) FIG. 11B shows another lens design to create a lens 100 that achieves at least one of the three inequalities (4)-(6) in another manner. In this design:
.sub.nvr=.sub.dvr.(8)

(81) Instead of modifying the surface tangentials, in these embodiments the distance-vision region 110 has a distance-vision z-axis; the near-vision region 120 has a near-vision z-axis, and the near-vision z-axis is rotated, or twisted in a nasal direction relative to the distance-vision z-axis. The twist is illustrated from looking down on the lens from they axis direction. The distance-vision z-axis at the highest y levels of the lens 100 where the distance-vision region 120 is naturally located, can be essentially parallel to the overall lens z-axis 3. Progressing towards lower y levels, where the near-vision region 120 is naturally located, the x-y plane of the lens is getting rotated so that the z-axis is rotated in the nasal direction. Two of the twisted cross sections are shown in FIG. 11B. The middle cross section may correspond to the progression region 130, and the bottom, most twisted cross section can correspond to the near-vision region 120, with its twisted near-vision z-axis.

(82) It is noted that a manufacturing process of the embodiment of FIG. 11B may be remarkably easy as the process may involve forming a lens with the desired optical powers and then warming the lens until its material softens to a degree that allows a twisting of the lens by the designed amount.

(83) Next, the embodiments of FIGS. 7A-D will be described. FIG. 7A shows that embodiments of the convergence-reducing lens 100 can include a distance-vision region 110, having a non-negative distance-vision optical power, that is configured to refract a light ray 2, directed parallel to the z-axis 3 at a distance-vision region point P.sub.d at an x-distance from a y-z plane of the coordinate system x.sub.Pd, to intersect the y-z-plane at a distance-vision intersection z-distance z.sub.Id. The convergence-reducing lens 100 can also include a near-vision region 120, having a near-vision optical power that matches the distance-vision optical power within 0.5 D, that is configured to refract a light ray 2, directed parallel to the z-axis 3 at a near-vision region point P.sub.n, at an x-distance x.sub.Pn that is equal to the distance-vision region point P.sub.d: x.sub.Pn=x.sub.Pd, to intersect the y-z-plane at a near-vision intersection z-distance z.sub.In that is greater than the distance-vision intersection z-distance:
z.sub.In>z.sub.Id.(9)

(84) In some embodiments of the convergence-reducing lens 100, the distance-vision region 110 can be configured to refract the light ray 2, directed parallel to the z-axis 3 at the distance-vision region point P.sub.d at the x-distance x.sub.Pd, by a distance-vision refraction angle .sub.d. The near-vision region 120 can be configured to refract the light ray 2, directed parallel to the z-axis 3 at the near-vision region point P.sub.n at the x-distance x.sub.Pn, by a near-vision refraction angle .sub.n. In embodiments, .sub.n.sup.x, an x-component of the near-vision refraction angle .sub.n can be smaller than .sub.d.sup.x, an x-component of the distance-vision refraction angle ad that corresponds to the same x-distance x.sub.Pn=x.sub.Pd:
.sub.n.sup.x<.sub.d.sup.x.(10)

(85) In some embodiments of the convergence-reducing lens 100, the distance-vision region 110 can be configured to refract the light ray 2, directed parallel to the z-axis 3 at the distance-vision region point P.sub.d at the x-distance x.sub.Pd, to intersect the y-z plane with a distance-vision gaze-convergence angle .sub.d; the near-vision region 120 can be configured to refract the light ray 2 directed parallel to the z-axis 3 at the near-vision region point P.sub.n at the same x-distance x.sub.Pn=x.sub.Pd, to intersect the y-z plane with a near-vision gaze-convergence angle .sub.n. In embodiments, the near-vision gaze-convergence angle .sub.n can be smaller than the distance-vision gaze-convergence angle .sub.d that corresponds to the same x-distance:
.sub.n<.sub.d.(11)

(86) The inequalities (9)-(11) characterize the embodiments of FIGS. 7A-B similarly to the inequalities (4)-(6) characterizing the embodiments of FIGS. 5A-B and FIGS. 6A-B. Embodiments of the convergence-reducing lens 100 satisfy at least one of the three inequalities (9)-(11).

(87) As before, embodiments of the convergence-reducing lens 100 can further include a progression region 130, at least partially between the distance-vision region 110 and the near-vision region 120, that is configured to refract a light ray 2, directed parallel to the z-axis 3 at a progression region point P.sub.p at the x-distance x.sub.Pp that is the same as of the distance-vision region point x.sub.Pp=x.sub.Pn=x.sub.Pd, to intersect the y-z-plane at a progression intersection z-distance z.sub.Ip that is between the near-vision intersection z-distance z.sub.In and the distance-vision intersection z-distance z.sub.Id: z.sub.Id<z.sub.Ip<z.sub.In.

(88) FIGS. 7C-D describe embodiments that are related to reversing the path of the light rays 2 in the embodiments of FIGS. 7A-B, albeit with some necessary adaptations. FIG. 7C illustrates that embodiments of the convergence-reducing lens 100 can include a distance-vision region 110, having a non-negative distance-vision optical power, configured to refract a light ray 2, directed by a source 15r at a distance-vision region point P.sub.d at an x-distance x.sub.Pd from a y-z plane of the coordinate system, to make a distance-vision light-convergence angle 6d with the y-z-plane, wherein the source 15r is located on the z-axis 3 at an intersection z-distance z.sub.Id from a center of the coordinate system. The lens 100 can further include a near-vision region 120, having a near-vision optical power that matches the distance-vision optical power within 0.5 D, that is configured to refract a light ray 2, directed by the source 15r at a near-vision region point P.sub.n at the same x-distance x.sub.Pn from the y-z plane of the coordinate system as that of the distance-vision point P.sub.d: x.sub.Pn=x.sub.Pd, to make a near-vision light-convergence angle .sub.n with the y-z-plane. Here the source 15r can be at the intersection z-distance z.sub.In that is again the same as the source 15r for the distance-vision z.sub.Id: z.sub.In=z.sub.Id. In such embodiments, .sub.n.sup.x, an x-component of the near-vision light-convergence angle .sub.n can be greater than .sub.d.sup.x, an x-component of the distance-vision light-convergence angle .sub.d:
.sub.n.sup.x>.sub.d.sup.x.(12)

(89) In some embodiments of the lens 100, the distance-vision region 110 can be configured to refract the light ray 2, directed by the source 15r at the distance-vision region point P.sub.d at x.sub.Pd, the x-distance from the y-z plane of the coordinate system, by a distance-vision refraction angle .sub.d. Further, the near-vision region 120 can be configured to refract a light ray 2, directed by the source 15r at the near-vision region point P.sub.n at x.sub.Pn, the x-distance from the y-z plane of the coordinate system, by a near-vision refraction angle .sub.n. In embodiments, .sub.n.sup.x, an x-component of the near-vision refraction angle .sub.n can be smaller than .sub.d.sup.x, an x-component of the distance-vision refraction angle .sub.d:
.sub.n.sup.x <.sub.d.sup.x.(13)

(90) Inequalities (12)-(13) characterize the embodiments of FIGS. 7C-D analogously to inequalities (4)-(6) characterizing the embodiments of FIGS. 5C-D, and inequalities (9)-(11) characterizing the embodiments of FIGS. 7A-B.

(91) Several additional characteristics of the embodiments of FIGS. 5A-D and FIGS. 6A-B were described earlier. These characteristics can also apply to, or combined with, the embodiments of FIGS. 7A-D.

(92) FIGS. 12A-B show embodiments of an eye-strain reducing lens 100, or convergence reducing lens 100. These embodiments can be characterized via a description of the curvatures of the lens surfaces and the off-center locations of their corresponding centers of curvatures. In some detail, embodiments of the eye-strain-reducing lens 100, or convergence-reducing lens 100 can have a central normal of the convergence-reducing lens that defines a z-axis 3. The z-axis 3 is typically also the z-axis of a distance-vision region 110. A central region of the convergence-reducing lens 100 can further define a tangential, centered x-y plane. The z-axis 3 and the x-y plane together define an x-y-z coordinate system.

(93) The convergence-reducing lens 100 can include the above mentioned distance-vision region 110 with a non-negative distance-vision optical power, having a front distance-vision surface 140df with a radius of curvature R.sub.df and a center of front distance-vision curvature CC.sub.df, and a rear distance-vision surface 140dr with a radius of curvature R.sub.dr and a center of rear distance-vision curvature CC.sub.dr. The lens 100 can further include a near-vision region 120 with an optical power within 0.5D of the distance-vision optical power, having a front near-vision surface 140nf with a radius of curvature R.sub.nf and a center of front near-vision curvature CC.sub.nf, and a rear near-vision surface 140nr with a radius of curvature R.sub.nr and a center of rear near-vision curvature CC.sub.nr; wherein an x-coordinate of the center of front near-vision curvature x(CC.sub.nf) can be nasal relative to an x-coordinate of the center of front distance-vision curvature x(CC.sub.df), or an x-coordinate of the center of rear near-vision curvature x(CC.sub.nr) can be temporal relative to an x-coordinate of the center of rear distance-vision curvature x(CC.sub.dr). Expressed the above attributes in inequalities, and using the directionality of the x-axis, such that points lying to the right, temporal direction have greater x coordinates than points lying to the left, nasal direction, these conditions can be written as:
x(CC.sub.nf)<x(CC.sub.df), or(14)
x(CC.sub.nr)>x(CC.sub.dr).(15)

(94) In some typical embodiments, the CC.sub.df front and CC.sub.dr rear centers of curvature of the distance-vision surfaces 140df and 140dr can be located on the z-axis 3 and therefore, their x coordinates can be zero. In formal terms, x(CC.sub.df)=x(CC.sub.dr)=0. In such embodiments, the convergence-reducing lens 100 can be configured so that x(CC.sub.nf), the x-coordinate of the center of front near-vision curvature CC.sub.nf, is nasal relative to the z-axis 3 of the coordinate system, i.e.:
x(CC.sub.nf)<0, or(16)

(95) x(CC.sub.nr), the x-coordinate of the center of rear near-vision curvature is temporal relative to the z-axis 3 of the coordinate system, i.e.
x(CC.sub.m)>0.(17)

(96) In this sense, embodiments of the convergence-reducing lens 100 are off-axis center of curvature lenses.

(97) The above-described coordinates and x-distances of the centers of curvature x(CC.sub.nf), x(CC.sub.nr), x(CC.sub.df), and x(CC.sub.dr) can be determined with specialized tools and devices, such as spherometers and lens profilometers.

(98) Designs of the convergence-reducing lens 100 can achieve the optical power of the near-vision region 120 to match the optical power of the distance-vision region 110 within 0.5 D because the optical power in first approximation is given by the radii of curvature of the lens front and rear surfaces: Optical power (distance-vision)=f(R.sub.df, R.sub.dr), and Optical power (near-vision)=f(R.sub.nf, R.sub.nr). In the thin lens approximation, the optical power is proportional to f(R.sub.1, R.sub.2)=(n1)(1/R.sub.11/R.sub.2). As long as f(R.sub.nf, R.sub.nr)=f(R.sub.df, R.sub.dr), the optical powers in the two regions are matching in a leading order approximation.

(99) However the above relations assume that the centers of curvatures are on the main optical axis of the lens. So, designs of the lens 100 can be viewed as built on the recognition that it is possible to leave the optical power of the near-vision region 120 essentially equal to that of the optical power of the distance-vision region 110 by not manipulating the radii of the corresponding curvatures, yet, to adjust and manipulate the near-vision refraction angles relative to the distance-vision refraction angles by moving the centers of curvature off the axis of the lens. More concisely, in designs of the lens 100 it is possible to make the refraction angles of the near-vision region different from the refraction angles of the distance-vision region, while preserving that the optical power of the near-vision region matches the optical power of the distance-vision region. The refraction angles and the optical powers of these two regions are adjustable relatively independently from each other.

(100) Some embodiments of these convergence-reducing lenses 100 can be further characterized as follows. With reference to FIG. 11A, the front distance-vision surface 140df and the rear distance-vision surface 140dr, at an x-distance from a center of the coordinate system, can define a distance-vision surface convergence angle .sub.dvr; and the front near-vision surface 140nf and the rear near-vision surface 140nr at the same x-distance from the center of the coordinate system can define a near-vision surface convergence angle .sub.nvr, wherein in embodiments the near-vision surface convergence angle is smaller than the distance-vision surface convergence angle:
.sub.nvr<.sub.dvr.(18)

(101) The convergence-reducing, off-axis curvature center lenses 100 can be further characterized by, and combined with, the features described in relation to FIGS. 1-11.

(102) While this document contains many specifics, these should not be construed as Limitations on the scope of an invention or of what may be claimed, but rather as descriptions of features specific to particular embodiments of the invention. Certain features that are described in this document in the context of separate embodiments can also be implemented in combination in a single embodiment. Conversely, various features that are described in the context of a single embodiment can also be implemented in multiple embodiments separately or in any suitable subcombination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a subcombination or a variation of a subcombination.