User:Jkopriva/sandbox

Polyfocality and polyfocal lenses
Simple optical lenses are usually expected to be monofocal. Monofocal lens is a device capable of focusing paraxial incoming rays of light into a single spot called “focal point” on the optical axis [Fig.1]. Strictly speaking, not many lenses are truly monofocal: only a lens with parabolic optical surface focuses all paraxial rays into one single point, no matter whether they are passing through the center of the lens or its periphery.

The most of the common lenses have spherical surfaces that can be manufactured rather readily. However, spherical lens demonstrates so called “spherical aberration”: rays incoming through the center are bent into a focal point that is slightly further from the lens than rays incoming through the lens periphery [Fig. 2]. The lens is becoming more refractive from the center to the periphery. Such a lens does not have a single focal point, but many focal points in a short interval of distances for rays going through the lens at different distances from the optical axis. The spherical lens is just slightly polyfocal, although its focal range is rather small and besides, it goes “in a wrong way for accommodatin” as will be shown later. Example of local refractive powers of a spherical lens vs the distance from the center is in Fig. 3. (Refractive power in Diopters [Dpt] is reciprocal of focal distance in meters [m]). Lenses with elliptical rather than spherical surface (such as surfaces created by solidification of a static liquid meniscus) have even much more distinct spherical aberration.

Lenses with at least one hyperbolic surface demonstrate a “hyperbolic aberration” that is opposite of the spherical aberration: rays incoming through the center are bent into a focal point that is closest to the lens, and the focal point is getting progressively further from the lens for rays incoming in increasing distance from the lens center toward the lens periphery [Fig. 4].

The lens with hyperbolic surface has highest refractive power at its center, and the refractive power decreases from the center toward the lens periphery. The focal range of a hyperbolic lens can be rather large and is controllable by so called conic constant or shape parameter defining the hyperbolic surface shape.

Y – Yo = [Ro(X-Xo)^2]/{1+[1-hRo^2(X-Xo)^2]^0.5}……………………………………..[1]

where Ro is the central radius of curvature and h is the conic constant (or shape parameter). The Eq. [1] describes any conic section curve – parabola, circle, hyperbole, etc. depending on the shape parameter h value. Value of h for hyperbolic surfaces is h< 0. Distribution of refractive power in a lens with hyperbolic surface is shown in [Figure 5]. As established in recent years [ Ref. 1,2,3 ], both anterior and posterior surfaces of a young human lens are hyperbolic. It was found that for a typical young human the anterior surface of the natural lens is more hyperbolic than the posterior surface, that hyperbolicity increases significantly with accommodation, and that the human lens grows with age and its hyperbolicity decreaeses so than an old lens may become substantially spherical. According to these references, a typical human lens anterior central radius ranges from about Ro = and the average anterior conic parameter about  h = – 4 (ranging from about – 22 to to +6). The posterior central radius ranges from about Ro = and the average posterior conic parameter is about h – 3 (ranging from about -14 to + 3). Therefore, natural human lens (and particularly young human lens) is distinctly polyfocal and its polyfocality further increases by the accommodation effort. The are also artificial polyfocal lenses with distinct hyperbolic aberration on the market, notably bioanalogic WIOL-CF manufactured and distributed by MEDICEM Group.

Polyfocality of the natural lens helps the eye accommodation in several ways:
 * 1) It projects on retina simultaneously images of objects in all distances covered by the focal range of the lens; this essentially increases the depth of the focus of the eye.
 * 2) It increases hyperbolicity of the lens by the accommodation, which further increases the focal range of the lens ant, therefore, the depth of the focus still further;
 * 3) The eye helps to focus on near objects by narrowing the pupil. This so called “pupillary reflex” or “near mydriasis” has two consequences: first, the decreases aperture and thus increases depth of the focus of the eye as the optical system (narrowing aperture blocks rays that are far from the axis and coming in sharp angles with respect to the axis); and it increases the average refractive power of the lens by using only its central portion with the highest refractive power.  Effect of the near mydriasis is schematically shown in [Figure 6].

It is obvious that near mydriasis assist the near focus only for lenses with hyperbolic aberration. It has little effect in parabolic lenses, and in spheric or even eliptic lenses it has entirely opposite effect (spherical or meniscoid lens becomes by near mydriasis weaker lens with lower refractive power, so that it is adjusted for distant focus, rather than lens with higher refractive power needed for near focus).

Let me try to explain the visual effect of polyfocality resulting from a hyperbolic aberration in the following way:

The optical system with a monofocal lens forms the image of an object in one specific distance that is placed in one specific image plane. If a device for the image visualization (such as a screen, a camera chip or retina) is placed exactly in the image plane, a sharp, focused image is recorded. If the device is placed nearer or further from the lens than the image plane, then the recorded image becomes dis -focused in proportion with the distance between the image plane and the visualization device. Therefore, a monofocal lens system with fixed lens and device can visualize sharp images of object in one and only one distance whereas images of objects in all other distances are more or less disfocussed.

A bifocal lens has two parts, each of them being “monofocal”. Rays incoming through each part form two different images of each object at two different image planes. The refractive powers of each part are selected in such a way that image plane for one part coincides with the position of viewing device for far objects and other for near objects. Images of all objects between the two preselected distances (e.g., near and infinity) are projected in front of or behind the viewing device, and their image on the viewing device is dis-focused. Similar situation would apply to trifocal lenses, or generally to multifocal lenses composed from a finite number of monofocal parts.

In a polyfocal hyperbolic system, there are many (actually, an infinite number of) images formed for every object, no matter what is their distance. If the focal range is properly selected, there will be image plane coinciding with the viewing device for any object in the field of view. In other words, there will be always a sharp image projected on the viewing device for all objects which distance corresponds to the focal range of the polyfocal range. (Focal range is the interval of focal distances from the nearest one corresponding to the lens center to the farthest one corresponding to outside rim of the lens, which is defined by the pupil aperture. To this corresponds the depth of focus – the focal range should be selected such that all objects at distance from, say, 30 cm to infinity could form a sharp image on retina).

The focal range can be further extended by change in the pupil aperture (near mydriasis), change of the lens shape (decrease of the radius of curvature and shape parameter), change in lens-to-retina distance (A-P lens movement and/or scleral extension that are sometimes assumed but rarely observed) and increase of the lens refractive index. Human eye may use all these mechanism, but it is not clear if every eye uses all of them in the same proportion and under all circumstances. In any case, the polyfocality appears to be the important part of the accommodation mechanism.

Both in the case of a monofocal and polyfocal lens, each sharp image on the viewing device is accompanied by a multitude of dis-focused images that decrease resolution and contrast sensitivity of human eye. In the case of the monofocal lens, these are images of all objects in the field of view that are nearer or further than the object with the sharply focused image. In the case of a polyfocal lens (whether natural or artificial), these are images of all objects from those areas of the polyfocal lens that cannot focus them correctly. Our eye/brain learned to cope with this image degradation by several possible mechanisms, such as:
 * 1) Each photosensitive element has a certain finite size, so that their activation may be very similar for perfectly focused and slightly dis-focused images.
 * 2) There is a certain threshold light intensity needed in order to be registered by the photosensitive element.  Very dis-focused images are spread over a wide area and over many photosensitive elements so they either do not register, or they add just marginally to the light intensity from the focused images on that particular element.
 * 3) Our brain learned to suppress dis-focused images to obtain resolution and contrast sensitivity that is functional for humans (though they could be insufficient for hawks or eagles).