Light Sword in the eye

There are various methods for presbyopia correction. One of them are mono-, bifocal or progressive prescription glasses, or mono- or multifocal contact lenses. Depending on the design of the element, they provide good contrast and create sharp, fairly bright images of visible objects in certain distances. However, their applicability is somewhat limited. Reading glasses work well for near objects, but are unsuitable for watching television or commuting.
The other solution for presbyopia correction is application of the imaging methods with extended depth of field, which means providing acceptable sharpness of vision in certain distances. In other words, to replace the lost eye accommodation ability.

Fig. 1:  Schemes showing differences between small and large depth of field. In the case of imaging system with small depth of field only the object, lying in the best sharpness plane, is in focus. Images of other objects are blurred. In the case of the imaging system with large depth of field, all considered here objects are in focus – their images are sharp.

Fig. 2: Scheme explaining the Circle of Confusion concept. Distances "v" denote the positions of the object images. Index "f" denotes the image behind the focus plane and "n" in front of the focus plane, C – Circle of Confusion.

The concept of the depth of field is rather intuitive, but certain clarification is necessary.  Optical imaging systems have strictly defined plane, in which the image of the object is reconstructed. In ideal optical system, the sharp image can be observed only in this plane. However, what we see as a “sharp” image is not automatically equal to the definition of the sharp image stated above. It can be explained using the quantity called Circle of Confusion (CoC). It is the maximum diameter of the disc, which human eye cannot distinguish from the point source of light at given distance (see fig. 2). Its value depends on set parameters like: distance, from which we observe the final image (e.g. photograph), the magnification of the final image in respect to original, vision sharpness. This means that objects that are not particularly in the focus plane can also be seen as sharp.

Having in mind what was said above, the depth of field is range of distances, in which the spreading of the points in the final image is smaller than the Circle of Confusion, and this means the range of distances in which the objects in the image are seen as sharp.
Term depth of field corresponds to the object space (where the objects are) and refers to the range of the object distances in which objects are in focus in the image. However, the term depth of focus corresponds to the image space (where the observation plane is) and refers to the range of image distances in which the images of the objects are sharp. Thus, it is more natural and convenient to use the term depth of field in this case, because it is related to the real world we live in and not its image.

Fig. 3: Examples of the images with small and large depth of field. 

We can distinguish two groups of the imaging methods with extended depth of field

• Opto-digital
• Optical

Opto-digital methods use special optical system design coupled with following digital post-processing of the obtained image. One of this methods is Wave Front Coding (WFC) proposed by E.R Dowski Jr. and W. Thomas Cathey in 1995. It uses a special optical element (phase plate), which modifies the properties of the optical system (lens + imaging sensor). We can say, that the imaging system becomes insusceptible to defocus (observed objects are not in the focus plane) in certain range of distances. The characteristic blurring of the image (coding), introduced by the phase plate (see fig. 4), is similar for the whole considered distance range.

Fig. 4: Example of the imaging results with system using Wave Front Coding method.

The mentioned blurring can make in some cases images hard to recognize, but it does not cause any information (detail) loss, in contrary to the traditional imaging system like photographic camera lens. Thus, it is possible to recover the image with extended depth of field, by using the digital post-processing.
In scientific literature, one can find a wide range of analysis and implementations of the imaging systems, which use Wave Front Coding. This method can be applied in the field of microbiology or infrared imaging.
Other method to solve the problem is to combine the set of images with small depth of field (focus stacking). It is used e.g. in macro photography. The concept of this approach is to obtain a series of images, changing the focus plane for each photograph. Then, by using proper algorithm, the images are combined into one with extended depth of field (fig. 5).

Fig. 5: Example of extended depth of field imaging by Focus stacking. Work by Muhammad Mahdi Karim, Source: https://en.wikipedia.org/wiki/Focus_stacking#/media/File:Focus_stacking_Tachinid_fly.jpg

The other interesting method is applied in so-called light-field camera (plenoptic camera). Its concept is based on integral (complete) photography proposed by Gabriel Lippmann. It uses a micro lens array placed in certain distance from the imaging sensor. Light, focused by each micro lens, is collected by assigned to it group of pixels. Each pixel from the group corresponds to certain light ray, specifying its position and incidence angle. Thus, by proper algorithms, one can modify the plane of focus location after the image is taken and also  extend the depth of focus.
Although, the opto-digital methods give many possibilities to enhance recorded images, they require the digital post-processing, which means that they cannot be used for presbyopia correction.
The optical methods use elements, which properties include imaging with extended depth of field and for this reason, they do not need digital post-processing. A few of optical methods are listed below:

• aperture decrease – decrease of the transparent area of the optical system
• application of the binary phase filters
• application of the holographically generated complex filters
• application of the optical elements like axicons and Light Sword Optical Element (LSOE)

In photography the extended depth of field can be obtained be decreasing the aperture of the lens. Similarly, the depth of field can be extended by limiting the transparent area of the contact lens to diameter of 2-3mm. However, it is also associated with significant reduction in light reaching eye, and thus leads to decrease of the vision quality, especially in low light conditions.
The second method uses the optical element, which extends the lens focus into the section along the optical axis. Third one is based on design of the diffraction element (holographic plate), which works similarly to lens, but focuses light into the section – creates the hologram of the luminescent section.
The last of the mentioned methods of imaging with extended depth of field uses special optical elements such as axicons and LSOEs. Just like the complex filters they focus light in the section, but in a different way.
Axicon is an axis-symmetric structure with continuous modulation of the optical power (or focal length) in radial direction. This means, that each thin ring of the element has different focal length.
The Light Sword Optical Element is a structure with continuous angular modulation of the optical power (focal length) in the range [0°; 360°). It is not axis – symmetric. This means, that each thin angular sector has different focal length.

Fig. 6: Source: J. Ares García, S. Bará, M. Gomez García, Z. Jaroszewicz, A. Kolodziejczyk, and K. Petelczyc, "Imaging with extended focal depth by means of the refractive light sword optical element," Opt. Express 16, 18371-18378 (2008)

The name of the element refers to the “light sabers” known from the Star Wars movie series (in polish the names both the lens and the weapon of the Jedi warriors are translated to the same phrase “miecz świetlny). Thin angular sector of the lens creates a section PP1 normal to the optical axis, with maximum of the intensity distribution lying off-axis between the points P and P1 . Each angular section creates such distribution in different place along z axis. The maxima of these intensity distributions form an focus spot extended in z direction. Thus, LSOE focuses light into the the certain focal section in space. One can say that such focus section looks like the sword with blade made of light.

One of the advantages of the LSOE over multifocal lenses is its relatively good imaging quality over the designed range. Multifocal lenses provide good imaging properties only for the distances corresponding to focal length of particular areas of the element. Moreover, the LSOE does not change its parameters (especially optical power range) with aperture diameter. In the case of axicons and multifocal lenses, the decreasing aperture diameter limits their active area, influencing the imaging quality in considered distance range.

Fig. 7: Sample image of the LSOE

However everything has its price, the LSOE will not provide the same contrast and sharpness as mono-focal lens. The same problem concerns both axicons and multifocal lenses. The reason for this is that the light reaching the lens is used for reconstruction of more than one image.
Nevertheless, the imaging properties of the Light Sword Optical Elements are very promising considering presbyopia correction.
The LSOE was originally developed and characterized in collaboration of Faculty of Physics Warsaw University of Technology, Facultade de Fisica, Grupo de Optica , Universidade de Santiago, and Central Laboratory of Optics (now: Institute of Applied Optics) by Andrzej Kołodziejczyk, Salvador Bará, Zbigniew Jaroszewicz and Maciej Sypek  in 1990.
The example of the LSOE performance is presented in fig. 8. The images were captured with artificial eye phantom. It is a device, which models the optical parameters of the human eye, especially vision geometry. The phantom included the color camera for image acquisition.

Fig. 8: Comparison of the presbyopic eye vison (left) and presbyopia correction (right) with LSOE. The images were obtained with artificial eye phantom, which focus plane was set at infinity

The results obtained so far show, that LSOE is a promising structure for presbyopia correction. Although this lens is not perfect in various aspects, the long list of advantages over current solutions encourage for its further development. The focus of the ongoing research is set on application of the LSOE as contact lenses or intraocular implants.
Who knows, maybe in near future we will have Light Sabers…ehh Swords in eyes.

The images in this article are part of the research conducted under EU project No. 315564 LightSWORDS, which SKA Polska has participated in..

References:

1. J. Conrad, „Depth of Field in Depth”, 2006, http://www.largeformatphotography.info/articles/DoFinDepth.pdf
2. J. Ares García, S. Bará, M. Gomez García, Z. Jaroszewicz, A. Kolodziejczyk, and K. Petelczyc, "Imaging with extended focal depth by means of the refractive light sword optical element," Opt. Express 16, 18371-18378 (2008)
3. Andrzej Kołodziejczyk , Salvador Bará , Zbigniew Jaroszewicz , Maciej Sypek, “The Light Sword Optical Element—a New Diffraction Structure with Extended Depth of Focus”, Journal of Modern Optics, Vol. 37, Iss. 8, 1990
4. Edward R. Dowski, Jr., W. Thomas Cathey, „Extended depth of field through wave-front coding”, Appl. Optics, 34, 1859 (1995)
5. S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, J. van der Gracht, „High-Resolution Imaging Using Integrated Optical Systems”, Wiley Periodicals, Inc., 14, 67 (2004)