Matrix Methods for Optical Layout

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Decomposition of matrices and system synthesis -- 1. Central theorem of first-order ray tracing -- 1. Aperture stop and field stop -- 1. Lagrange invariant -- 1. Petzval radius -- 1. Delano diagram -- 1. Phase space -- 1. An alternative paraxial calculation method -- 1.

Gaussian brackets. All rights reserved. Remember me on this computer. Cancel Forgot your password? Gerhard Kloos. Tutorial texts in optical engineering , TT Print book : English View all editions and formats. This book is intended to familiarize the reader with the method of Gaussian matrices and some related tools of optical design.

Optics -- Mathematics. View all subjects. Similar Items. Gerhard Kloos Find more information about: Gerhard Kloos. Optical components. Components based on reflection -- 2. Components based on refraction -- 2.


Components based on reflection and refraction. Sensitivities and tolerances. Cascading misaligned systems -- 3. Axial misalignment -- 3. Beam pointing error. Anamorphic optics. Two alternative matrix representations -- 4. Orthogonal and nonorthogonal anamorphic descriptions -- 4. Cascading -- 4. Rotation of an anamorphic component with respect to the optical axis -- 4.

Since the entrance pupil of the eye remains unchanged with accommodation and with IOL implantation, the input reference ray remains constant.

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A further advantage of the angular magnification is that it is independent of the axial length of the eye. Lateral magnification can be determined readily from the angular magnification given a value for the axial length. Our results suggest that both the angular and the lateral magnifications of the eye increase proportionally with translation of the lens elements during pseudophakic accommodation. Both a neural 36 and optical 35 cause for this effect have been proposed. A computational study demonstrated the optical minification effect of accommodation in a range of schematic eyes for which the retinal image size decreased by as much as 1.

We found a slightly higher value 2. The amount and direction of translation and powers of the lens elements are major design parameters significantly associated with magnification. These design parameters are also relevant to the accommodative performance of the AIOL. To achieve a given amount of accommodation, an AIOL with higher front element power requires less translation than an AIOL with lower front element powers 16 , With an equal amount of translation, a higher power of the front element produces a higher amount of magnification.

However, because accommodation is also proportional to the distance of translation of the optics, an AIOL with higher power of the front element will produce lesser magnification at a given accommodation amplitude. An optical explanation for the change in magnifications in translating-optics AIOLs can be based on the relative position of the image space nodal point. Retinal image height is dependent on the distance between the retinal plane and the image space nodal point.

For a given nodal ray angle, the image height increases as the nodal point to retina distance increases.

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In our model, the distance between the retina and the image space nodal point increased with translation of the optical element Figure 5 which led to an increase in magnification. The rate of change in distance between the image space nodal point and the retina per millimetre of translation was greater for a 2E-AIOL with higher power of the front element.

These results highlight a potential clinical issue relating to implantation of AIOL - that of dynamic aniseikonia. Dynamic aniseikonia may arise in two ways; either directly as a result of inter-ocular differences in magnification, or indirectly due to aniso-accommodation of AIOL between eyes. Assuming a nominal 2. Taking into account the accommodative minification of the normal phakic eye, this suggests monocular implantation of an AIOL may have a deleterious effect on binocular visual function.

Even when AIOLs are implanted binocularly, care must be taken to ensure that bilateral magnifications are matched, at least within inter-ocular tolerances, along the entire range of accommodation in order to avoid dynamic aniseikonia-induced binocular imbalance. Though the difference in magnification between various design combinations of 2E-AIOL elements powers are reasonably similar for matching accommodation levels up to 4 D, dynamic aniseikonia may arise due to differences in accommodation between the two AIOLs i.

It must be appreciated that this scenario differs from the results of a study where 1E-AIOLs were implanted in one eye, and a conventional monofocal non-accommodating IOL was implanted in the other eye. The magnification introduced by the reading spectacles apparently compensated for the difference in magnification of the two types of IOLs. In order to facilitate modelling, some assumptions and approximations were used in the present study. Hence, there may be aspects which have not been considered. Since the thin-lens approximation was used, magnification produced by changes in the shape-factor of lens elements could not be evaluated.

Departure from constant magnification across the field distortion can be caused by implementation of specific individual or combinations of lens forms shape factors for the lens elements. Thus, the present results are theoretically only valid within the domain of paraxial optics approximations. However, since visual performance is typically primarily concerned with improving visual acuity, i.

The refractive status of the eye myopic or hyperopic and other ocular parameters such as corneal power and axial length, have also not been considered, but presumably have some effect on ocular magnification For a more precise prediction of the magnification effect for different refractive errors and ocular anatomical dimensions, analyses incorporating additional parameters would be required. In conclusion, from the optical design perspective of AIOLs, the power combination of the AIOL elements and the amount of translation are shown to be two key design parameters governing dynamic accommodation in the eye implanted with AIOL although implant position, refractive status and ocular dimensions should not be ignored.

The power combination of 2E-AIOL lens elements requires special attention to avoid potential dynamic anisometropia and dynamic aniseikonia. In some cases, customised selection of designs and element combinations may be required to match their performance in a binocularly balanced manner. We acknowledge scientific writer Dr. Judith Flanagan for editorial assistance. The accommodation-dependent model eye of Navarro et al 57 was used in the study as the reference phakic eye.

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Parameters marked with an asterisk vary with the amplitude of accommodation i. National Center for Biotechnology Information , U. Ophthalmic Physiol Opt. Author manuscript; available in PMC Jan 1. Author information Copyright and License information Disclaimer. Copyright notice. The publisher's final edited version of this article is available at Ophthalmic Physiol Opt.

See other articles in PMC that cite the published article. Abstract Purpose Using an analytical approach of paraxial optics, we evaluated the magnification of a model eye implanted with single-element 1E and dual-element 2E translating-optics accommodative intraocular lenses AIOL with an objective of understanding key control parameters relevant to their design. Methods Lateral and angular magnifications in a pseudophakic model eye were analyzed using the matrix method of paraxial optics.

Results Both lateral and angular magnifications increased with increased power of the front element in 2E-AIOL and amplitude of accommodation. Conclusion Power of the mobile element, and amount and direction of the translation or the achieved accommodative amplitude are important parameters in determining the magnifications of the AIOLs. Introduction Implantation of an intraocular lens IOL has become standard procedure for vision correction following cataract removal.

Methods Matrix ray-tracing The matrix method has been widely used in visual optics to calculate retinal image size, 17 spectacle and relative spectacle magnification 18 , 19 , and IOL power Lateral magnification Lateral magnification LM is defined as the ratio of the object size to image size Relative lateral magnification In addition to absolute lateral magnification, relative lateral magnification, analogous to relative spectacle magnification RSM , was calculated using two different approaches.

Angular magnification Various definitions for angular magnification are extant in the literature Open in a separate window. Figure 1.

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  • Image nodal point position During pseudophakic accommodation, positions of the cardinal points of the eye change due to movement of the AIOL. Computations For computational purposes, the finite model eye of Navarro et al 34 was used. Results Effect of Accommodation Both lateral and angular magnifications increased with accommodation in the pseudophakic eye whereas they decreased in the phakic model eye.

    Table 1 Changes in relative magnifications, angular magnification and image space nodal point N 2 position as a function of accommodation and translation in phakic and pseudophakic model eyes. Figure 2. Figure 3. Figure 4. Figure 5.

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    Figure 6. Figure 7. Discussion In this study, changes in both angular and lateral magnification of AIOLs were considered. Appendix The accommodation-dependent model eye of Navarro et al 57 was used in the study as the reference phakic eye. References 1. A new intraocular lens design to reduce spherical aberration of pseudophakic eyes. J Refract Surg. Wang L, Koch DD.

    Custom optimization of intraocular lens asphericity. J Cataract Refract Surg. Leyland M, Pringle E. Multifocal versus monofocal intraocular lenses after cataract extraction. Cochrane Database Syst Rev. Objective and subjective evaluation of photic phenomena after monofocal and multifocal intraocular lens implantation. Glasser A.

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    Restoration of accommodation: Surgical options for correction of presbyopia. Clin Exp Optom. Accommodative intraocular lenses: Considerations on use, function and design. Curr Opin Ophthalmol. Leaming DV. Heinz, J. Artman and S. The present invention, however, is more specifically concerned with a means and method for performing matrix-vector multiplication procedures in real-time, with the use of an incoherent light source, and with techniques appropriate to the use and processing of such incoherent light energy.

    The concept of the present invention employing an incoherent light source and optical techniques is inherently limited to the performance of matrix-vector multiplication operations when the column vector is real-positive, although the matrix may be complex.

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    • As a consequence, the concept of the present invention does not extend its usefulness to some versatile applications as a coherent optical processor, but it is nonetheless well suited for many real-time signal processing applications involving a broad variety of discrete linear transformation requirements. The embodiment and system illustrated in FIG. A condensing lens 11 receives the light energy output of the source 10 for transmission through an optical mask The light energy transmitted through the mask 12 is received by a scanning mirror 13, whence it is transmitted to an imaging lens The imaging lens 14, in turn, transmits the scanned light energy to a photo-responsive integrating detector The condensing lens 11 maximizes the light energy throughput in the system by imaging the light emitting from the light source 10 into the entrance pupil of the imaging lens The mask 12 may comprise a photographic optical transparency and, in the system illustrated in FIG.

      The scanning mirror 13 which is interposed in the optical path between the mask 12 and the imaging lens 14 causes the image of the light energy transmitted by the mask 12 to be repetitively translated with a constant velocity across the face of the detector As a result, the output column vector information is generated at the integrating detector In a typical embodiment of the system illustrated in FIG. In a typical embodiment as illustrated in FIG.

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      Within the concept of the present invention, a fundamental relationship is expressed by the imaging equation which relates to the exposure at the plane of the detector 15 in terms of the irradiance of a light field incident on the mask 12 and the intensity transmittance of the mask The quantity m t represents the transverse magnification between the plane of the mask 12 and the plane of the detector The velocity at which the image of the transparency is scanned across the face of the detector 15 is given by v.

      It can be shown that the output vector information is contained in the function E x',y'. The vector information may be in the form of an electrical time sequence of rectangular pulses as represented by FIG. Ideally, the light source 10 and the condensing lens 11 are configured in a manner such that the light energy incident on the mask 12 has a uniform irradiance distribution which varies in time according to the equation: EQU4.

      The rectangle functions appearing in equation 3 above may be expressed as EQU5. As was previously mentioned, FIG. Those skilled and knowledgeable in the art will appreciate that the present optical configuration as illustrated in FIG. The matrix information may be encoded on a photographic type optical transparency in binary or analog form. For ease of explanation and understanding, the present discussion is limited to the case in which the elements of the matrix are real-positive.

      However, the more general case of a complex matrix may readily be demonstrated and explained, though it is somewhat more complicated. The intensity transmittance of the matrix in the form of a photographic optical transparency representing binary encoding may be specified according to the equation: EQU6 The matrix contains a total number of MN clear rectangular apertures arranged in a rectangular array wherein each element has substantially the same area as is shown schematically in FIG.

      Accordingly, there is a one-to-one correspondence between each rectangular aperture or element in the array of the matrix in the form of a photographic optical transparency and each element in the mathematical matrix. The linear dimensions of the m th, nth aperture in the array are given by W and a mn in the x and y directions, respectively.

      W is the same for all such apertures, whereas a mn is equal to the mth, nth element of the matrix. The use of binary optical transparencies avoids many of the problems encountered in fabricating continuous tone analog masks; for example, binary masks have previously been employed in both holographic and coherent optical data processing systems as disclosed in publications by B. Brown and A. Lohmann and D. The result therefore, is EQU9.

      The foregoing mathematical expressions and procedures relate to an optical mask of the type illustrated in FIG. Any such discrete area, T mn is equal to KA mn , and by using the relationship, EQU10 and following essentially the same mathematical substitutions and procedures as described hereinbefore, the solution using the alternative type of mask may be obtained.

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      Matrix Methods for Optical Layout Matrix Methods for Optical Layout
      Matrix Methods for Optical Layout Matrix Methods for Optical Layout
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