Optical isolation of portions of a wave front

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1 2530 J. Opt. Soc. Am. A/ Vol. 15, No. 9/ September 1998 Charles Campbell Optical isolation of portions of a wave front Charles Campbell* Humphrey Systems, 2992 Alvarado Street, San Leandro, California Received December 8, 1997; revised manuscript received April 22, 1998; accepted May 1, 1998 A criterion is established for determining when portions of a wave front can be said to be optically isolated from the rest of the wave front in the sense that they can subsequently be treated separately when one is considering the formation of images. The subarea of the wave front is treated as a separate aperture, and it is said to be isolated if diffraction maxima for the majority of the wave front fall at or beyond the first minima for the subarea. An illustrative example employing two circular unequal-diameter apertures is presented. A method is given for identifying portions of wave front that may be optically isolated; the method uses the technique of fitting a reference surface to the actual wave front and then finding what is defined as the differential deflection of the actual surface with respect to the reference surface at all locations. Subpopulations of locations with similar differential deflection values, sufficient numbers, and sufficient differential deflection are candidates for area of optical isolation Optical Society of America [S (98) ] OCIS code: INTRODUCTION This paper addresses the question of when two portions of an optical wave front may be considered to be optically isolated in the sense that effects of one area do not appreciably influence the diffractive effects of the other in the formation of an image such that the two areas may safely be treated as separate optical elements for the purpose of investigating image formation. The motivation for such considerations arises from the field of vision, in which situations created primarily by irregularities in the corneal surface or by the effects to the crystalline lens caused by postsubcapsular cataracts lead to formation of double and even multiple images on the retina and hence cause monocular diplopia and various forms of so-called ghost images. The extensive treatment of aberrations developed over the years in the general field of optics does not truly address this situation, as optical instruments and optical systems are always designed to minimie aberrations, so the particular type of irregularity that leads to monocular diplopia in vision is not truly of concern to most of those involved in optics. It is true that there is a great body of work addressing the formation of images in the presence of aberrations. But here the emphasis is on the variations in the image caused by the transformation of the object by the optical system. The aberrations of the optical system certainly cause a less than faithful reproduction of the object in its image, and these effects have been well studied, but rarely do studied aberrations cause the optical system to act as if it were truly two or more systems. This paper proposes one way of predicting whether the aberrations found to be caused by an optical surface or system effectively isolate one portion of the created wave front from the rest of the wave front such that more than one image can be said to be formed. 2. DEVELOPMENT OF THE CRITERION FOR OPTICAL ISOLATION To investigate this question, a line of thought will be used that is similar to that used by Rayleigh 1,2 to set forth his celebrated criterion for the resolution of two point objects. In the Rayleigh criterion it is assumed that a single optical system exists and that an exit aperture can be defined in such a way that the diffraction pattern caused by it is known. This pattern may be considered to be the pointspread function of the optical system so that the effect of aberrations of the system can included if they are not too severe. The important thing is that the point spread shall have a principal maximum and shall have a series of amplitude eros (or minima) at various radial positions away from this principal maximum. The criterion for the resolution of two point objects of equal intensity imaged by such a system is that the principal maximum of one image shall fall no closer to the principal maximum of the other image than the distance of a principal maximum to its first amplitude ero. An implicit assumption in this analysis is that the investigation of the diffraction amplitude patterns take place in an image plane or close to it, i.e., a plane of high energy concentration, as it makes little sense to speak of resolution of images that are grossly out of focus. As there is only one exit aperture for the system, this distance is assumed to be the same for both images if an image can be said to truly exist at all. As aberrations are introduced into an optical system, the existence of true amplitude eros may disappear, but, as long as there is a sensible minimum in the vicinity of a principal maximum, the criterion is a good and useful one. Again, the Rayleigh criterion assumes a single optical system with multiple point sources of equal intensity, and it is assumed that diffraction amplitudes are investigated in an image plane or close to one. Now assume that there is only one point source but that the exit aperture is divided. We can always, in our treatment of the diffractive effects of a wave front emerging from an aperture, break the integration over the aperture into parts, because of the linear nature of the mathematics involved, and consider the total effect to be the sum of the contributions of these various parts. Naturally this can be done only before the amplitude and /98/ $ Optical Society of America

2 Charles Campbell Vol. 15, No. 9/ September 1998/ J. Opt. Soc. Am. A 2531 its complex conjugate are combined to yield the intensity, but this will serve for our purposes. A criterion, similar to the Rayleigh criterion, will now be imposed to see whether two portions of the exit aperture can be said to form two images. This criterion will be more complicated than the Rayleigh criterion because the subapertures are in general of different sie, whereas in the Rayleigh case there was only one sie to consider. Because of diffraction effects, a smaller aperture generally leads to greater distance between a central maximum and a first amplitude ero or minima. In addition, a smaller aperture leads to smaller peak value for the central maximum, with the amplitude maximum being proportional to the area of the aperture, so the central amplitudes are no longer equal. These two considerations suggest that, for the point spread created by a smaller subaperture to be resolved with respect to a larger subaperture, the diffraction amplitude maximum of the larger aperture should lie no closer to the diffraction amplitude maximum of the smaller aperture than the first diffraction amplitude minimum of the smaller aperture along a line connecting the two maxima. This ensures that the weak central maximum of the smaller aperture lies in a low-amplitude region of the larger aperture and can be distinguished. I suggest this condition as the criterion for the isolation of one part of a wave front from the rest of the wave front and say that, when this occurs, the two subapertures are optically isolated from each another for the purpose of image formation. In addition, I extend the concept of the aperture from one that considers only the exit aperture of a system to one that includes any continuous area that is essentially transverse to the direction of propagation of the optical disturbance. In particular, this view of the aperture allows powered surfaces to be included as apertures for the purpose of analysis. This is important, as it is they that create the optical conditions that give rise to optical isolation of portions of a wave front. 3. ILLUSTRATIVE EXAMPLES OF THE OPTICAL ISOLATION CRITERION To illustrate the effect of displacing the central maxima of two unequal apertures on the resulting point spread when the above criterion for optical isolation is satisfied, an example is now given in which circular apertures with no aberrations present are used. Physically this might be realied with a lens similar to a common bifocal spectacle lens with a round segment having the same power as the base lens, but with a prismatic offset. The case of circular apertures is chosen because of their widespread occurrence in optical systems (hence their general familiarity) and because the example may then be posed in one dimension owing to the radial symmetry of the apertures. In this way questions of orientation of the smaller aperture to the larger can be avoided. For the case of the circular apertures the diffraction amplitude functions, 3 A, are given by A iexpikexp i ka2 ka J 1 kar kar, A C ka2 2 2 J 1 kar kar, where a is the aperture radius, k 2/, is the wavelength, r is the radial position in the image plane, is the distance between the aperture and the image plane, C is a pure imaginary term that is independent of r. The ratio r/ will now be designated and used as the variable. Using this ratio as the variable casts the amplitude equation into angular space, so results can easily be correlated with a variable useful in visual science, i.e., prism deviation where the value in prism diopters,, is 100. The amplitude equation now takes the form A C a2 2 J 1 2a 2a. (1) Notice that the amplitude is scaled by the value a 2, which is the area of aperture. Since the various subapertures to be considered all have the same value of, the distance to the image plane, the ratio of central amplitudes is simply the ratio of aperture areas. Since the first ero of the Bessel function in Eq. (1) occurs when 2a 1.22, the criterion for optical isolation, in this case, is met when (2) a Note that a in Eq. (2) is the radius of the smaller of the two apertures considered, as has been specified in the suggested criterion for optical isolation. Figure 1(a) shows the individual amplitudes (shown by dashed curves) and their sum (shown by a solid curve) for two circular apertures whose diameter ratio is 1:2 (areas in the ratio 1:4) and whose central maxima are separated so as to fulfill the optical isolation criterion. Figure 1(b) shows the same two apertures, but here the separation is chosen to be the mean of the distance from central maximum to first minima for both apertures. This is a potential alternate criterion for optical isolation. The amplitude functions have been normalied so that the maximum of the amplitude sum has a value of 1.0. The larger aperture radius was chosen to be 2 mm, a value reasonable for visual science applications, and the wavelength, to be 500 nm. The deflection needed to satisfy the criterion for optical isolation in this case, given in prism diopters, is The deflection, expressed as an angular separation in minutes of arc, is 1.05 arc min. Those familiar with the capability of the visual system to resolve

3 2532 J. Opt. Soc. Am. A/ Vol. 15, No. 9/ September 1998 Charles Campbell Fig. 1. Amplitude and intensity plots for two circular apertures for two angular separation conditions. The larger aperture has four times the area of the smaller. (a) and (c) show the amplitude and intensity for the separation that fulfills the criterion for optical isolation, i.e., the first minima of the smaller aperture coincides with the maxima of the larger. The plots for the individual apertures are shown as dashed curves. The combined effect is shown as a solid curve. (b) and (d) show the condition for the same two apertures with a lesser separation, one equal to the mean distance to first minima for the two apertures. detail will recognie this value as being approximately equivalent to the stroke width of a 20/20 letter. The peak intensity for the secondary peak, representing the smaller aperture, is 9.8% of the primary maximum when the criterion for separation is satisfied. In the second case, when only a lesser deflection criterion is satisfied, the secondary peak is still evident, its maximum intensity has dropped to 6.8%, and it is shifted considerably from the peak of the smaller aperture taken by itself, so it could be thought of as a slightly aliased image. As the separation decreases, the secondary peak rapidly decreases in intensity and moves more from the peak of the second aperture considered by itself. Naturally these examples are idealied cases in that the individual point spreads are diffraction limited, so the energy away from the central maxima is quite low, and the weaker peak is easier to discern. In actual cases, other aberrations that open the point spreads of both subapertures and make minima less distinct make the smaller aperture s effects even less discernible. So, while in this example it may appear that the criterion for optical isolation might be relaxed, in practice it does not seem that this would be wise. Equation (2), which gives the deflection separation necessary for optical isolation in this idealied case, may be taken as the minimum necessary for optical isolation, as it is the best that can occur and so can serve as an approximate rule for minimum necessary separation. 4. METHODS FOR IDENTIFYING AREAS OF OPTICAL ISOLATION It will be readily appreciated that this criterion for predicting optical isolation of portions of a wave front and the formation of multiple images will not prove to be useful unless guidance can be given on the selection of subapertures that lead to multiple images. There are an infinite number of ways that a given aperture or wave front could be divided into parts, and most divisions will yield nothing of interest because the image formed by the chosen

4 Charles Campbell Vol. 15, No. 9/ September 1998/ J. Opt. Soc. Am. A 2533 subaperture will be essentially coincident with that formed by the rest of the aperture. A method is needed to find those particular subapertures whose central maxima are sufficiently decentered from the diffraction maximum created by the rest of the aperture as to cause multiple image to be possible. To find such a method, attention is first directed to the location of the central maxima of a subaperture. A standard technique, used ubiquitously in diffraction theory for analying a wave front, is that in which one defines an aberration function by subtracting from the actual wave front a reference wave front, often a spherical surface, which best fits it and by treating the remainder as the aberration function. It is usually found that the center of curvature of such a reference surface well approximates the center of the diffraction amplitude maximum. An equally common approach is to use as the reference surface a sphere whose center of curvature is located on the optical axis of the system in the nominal focal plane of the wave front. Such a standardied reference wave front is, in general, tilted from the one first described, and when a wave front is analyed by such a technique the tilt of the actual wave front with respect to the standardied reference wave front produces nonero values for the lowest-order coefficients. Common decomposition techniques are Zernike decomposition 4 6 and decomposition into generalied polynomials. 5,7 This concept may applied to the present problem in the following way. A technique, such as Zernike decomposition, is applied to the entire aperture, and tilt coefficients are found. The same procedure is then applied to the subarea of interest. The difference between the tilt coefficients found for the majority of the aperture and those found in the subaperture is formed. This will be termed the differential deflection 8 of the subaperture and gives a measure of the angular separation between the amplitude maxima of the two areas. The angular distance of the subarea first minima from its central maxima is also found. If it is smaller than the angular separation of the central maxima, a second image may be said to have formed. With a slight variation this approach can be used not only to extract the value needed to apply the optical isolation criterion but also to identify areas that are candidates for investigation in the first place. The total wave front is fitted with a low-order reference surface, in the Zernike sense. At each sample point location on the surface, the surface normal of the fitted surface is found. This is easily done with the fitting coefficients found and with a knowledge of the form of the reference surface. At each sample point a calculation is done for the local surface normal of the actual wave front. This is best done with a finite-element approach involving the point and its nearest neighbors. The vectorial difference between the two normals is now found, and the transverse components of the difference are designated the differential deflection at the point. Once the differential deflection values are found for all points, they may be plotted on a twodimensional plot whose coordinates are the orthogonal components of differential deflection. Such a plot immediately reveals the information needed. When a wave front contains areas leading to multiple image formation, this condition will be seen as separate discrete populations of plotted points. Symmetrical aberrations, such a spherical aberration, by contrast, will yield dispersed differential deflection distributions with no local concentrations, other than centrally. Such differential deflection plots are direct analogs of common ray-trace spot patterns, and they convey much the same type of information contained in such patterns. If the sample points in the wave front divide it into equal areas, the number of points in a local concentration in a differential deflection plot or in a ray-trace spot pattern is an indication of the sie of the subaperture and hence gives some indication of the deflection necessary for the subaperture to be said to optically isolated. An especially convenient way to find a differential deflection distribution is through the use of a complete Zernike decomposition of the actual wave front. After Zernike fitting coefficients of high enough order to ensure faithful representation of the wave front are found, only those terms not considered to be in the group composing the reference wave front are retained. These retained coefficients are then used to form gradients of their associated Zernike terms, and the sum of these gradients is considered to define the differential deflection field. A plot of these gradient vectors gives the differential deflection distribution. When a local population of points is found that is separated from the majority of the points in the aperture, the difference in mean differential deflection between the local population and the origin of the plot can be taken as the angular separation of the central maximum of the majority of the aperture and the subaperture. This information, in itself, is insufficient to allow one to tell whether optical isolation has occurred, even if the number of points in a subpopulation is used to estimate aperture sie, because the width of the point-spread function of an aperture in a given direction is determined not only by the area of the aperture but also by the shape and orientation of the aperture. What can be done is to see whether the optical isolation criterion is satisfied for a simple symmetrical geometry, such as that given in the example above. If it is, the area that gave rise to the points in question is a candidate for an optically isolated subaperture. Then a more careful calculation of the subaperture point spread in the direction connecting the center of the subpopulation and the center of the main population is made. This information can then be used to find the first minima and finally to see whether optical isolation has occurred. 5. DISCUSSION This paper has addressed the question of when two portions of an optical wave front may be considered to be optically isolated in the sense that effects of one area do not appreciably influence the diffractive effects of the other in the formation of an image such that the two areas may safely be treated as separate optical elements for the purpose of investigating image formation. It has not been directly concerned with the formation of multiple images, but, since it is hard to imagine what optical isolation could mean if multiple images did not form as a result of it, the subject of multiple image formation is intimately

5 2534 J. Opt. Soc. Am. A/ Vol. 15, No. 9/ September 1998 Charles Campbell related to the topic. Yet optical isolation, as herein defined, is not a necessary condition for the formation of multiple images, although it is sufficient, and I will briefly touch on that subject at the end of this discussion. The concept of optical isolation is a concept of subareas that act independently in a certain sense from the rest of the wave front of which they are a part and so are in the unusual position of being able to be treated separately in a diffractive sense from the rest of the optical disturbance. Optical isolation of a subaperture of a wave front has been defined to occur when the diffraction amplitude maximum of a major portion of the wave front lies no closer to the diffraction amplitude maximum of the subaperture than the first diffraction amplitude minima of the subaperture along a line connecting the two maxima. To identify subapertures that may be optically isolated from the rest of the wave front, the concept of differential deflection of an area of a wave front from the deflection expected from a best-fit reference wave front is used. Each point in the wave front has associated with it differential deflection values that are first used to see whether there are subpopulations with similar differential values. Then, if such subpopulations exist, their mean differential deflection can be used to see whether they are candidates for an optically isolated area. Those that qualify are treated as subapertures, and, using their calculated spread functions, one can see whether the criterion for optical isolation is satisfied. The question may now be raised whether a wave front that suffers optical isolation to a portion of it at one optical surface of system will continue to have a portion isolated as it passes through the remainder of the system. In general, this question may be answered in the affirmative. For a subsequent portion of a system to undo this effect, it would have act to create the inverse of the effect. Such an act is itself an act of optical isolation, but one of a very precisely controlled nature. This is highly unlikely. Optical isolation can be considered to be a disordering of a wave front, so one might say that its entropy is raised, which makes it very difficult to undo the effect. In this sense the wave front can be thought of as having a history and that at a point in that history a bifurcation has taken place. For all practical purposes, the portion of a wave front that is optically isolated acts as if it were created by a separate source. However, this second source is an unusual one, as it may be considered to have a high coherence with the actual source producing the majority of the wave front. So it is not truly like the two separate incoherent sources that are often assumed when applying the Rayleigh criterion. I leave open at this time the interesting question of what may be said about the state of partial coherence of a wave front that has suffered optical isolation. This question of whether optical isolation produced at one surface remains through a system is of specific interest with regard to an eye with irregularities in the anterior corneal surface. The eye may be considered to be a five-element optical system whose elements are more or less centered. There are four refracting surfaces to consider: the anterior corneal surface, the posterior corneal surface, the anterior crystalline lens surface, and the posterior crystalline lens surface. In addition, there is a gradient-index refracting element, the interior of the crystalline lens. Although the lens may not be well centered with respect to the two corneal surfaces, it has a high degree of symmetry within itself and is well suited to create single images. Hence the lens is quite unlikely to undo the creation of secondary images created by the cornea because that could be considered tantamount to creating multiple images by itself, albeit in reverse of the sense created by the cornea. The fact that portions of a wave front optically isolated by the cornea remain isolated on the retina allows corneal information alone to be used to assess the creation of multiple images. This is useful, as direct measurements of the optical parameters of the internal elements of the eye are not easily made, whereas it is much easier to make corneal measurements. Note that in this treatment no mention is made of the location of the subaperture within the full aperture. This is because only the mean differential deflection value of the subaperture influences creation of optical isolation. By contrast, the orientation of the subaperture relative to the direction of the differential deflection is important because it determines the distance between the main aperture central diffraction maximum and the first minima of the subaperture. As is the case for the Rayleigh resolution criterion, this criterion for optical isolation is assumed to be applied in an image plane or close to one. It is the use of a reference surface and differential deflection with respect to that surface that allows one to avoid directly specifying the image plane location. As can be appreciated from the example given, the ratio of the peak intensity of a secondary aperture to a primary aperture decreases quite rapidly as the areas become more and more dissimilar. This in turn means that the visibility of a secondary image in the presence of a primary image, if visibility is taken to be a function of contrast, decreases rapidly in a similar manner. So, while one can say that a portion of a wave front created by a small aperture is optically isolated based on the criterion here given, it may not create a visible second image not only because of the characteristics of the image detection system but also because of the characteristics of the object scene taken as a whole. 7 A full discussion of this aspect of secondary image formation is beyond the scope of this paper. Whether a secondary image, once optically isolated, can be detected or seen is an altogether separate question. This paper has considered optical isolation in terms of multiple diffraction peaks separated in planes transverse to the optical axis. Yet the concept could be enlarged with little difficulty to include images separated along the axis and would then include such interesting optical devices as certain types of bifocal contact and intraocular lenses. In addition, certain centered irregularities in the corneal surface that result today from refractive surgery also fall into this extended definition. As with the definition of optical isolation covered in this paper, optical isolation in an axial sense can be thought of as starting to occur when the first axial minima of the smaller subaperture coincides with the maxima of the larger subaperture. The existence of optical isolation of a portion of an ap-

6 Charles Campbell Vol. 15, No. 9/ September 1998/ J. Opt. Soc. Am. A 2535 erture is certainly a sufficient condition for the formation of a secondary image, but it is by no means a necessary one. There are other optical conditions that lead to the formation of multiple images. There are at least two global conditions, global in the sense that the effect is not localied to one portion of the aperture, which cause multiple images to form. A combination of the monochromatic aberrations spherical aberration and astigmatism will cause a doubling of the point-spread function in the vicinity of the astigmatic interval and so will cause doubling of images. 9 There is no localiation of the aberrations responsible in this case, so it cannot be considered to be a case of optical isolation. A second possible way to create double images is to place a phase grating in the incident and essentially collimated beam, which is designed to suppress even orders (hence the eroth order) while enhancing the 1 and 1 orders. One can do by introducing a half-wave delay between the grating stripes. The grating separates the incident beam in angle space into two beams, which are then brought to a doubled point spread in the image plane. Again, there is no localiation of the aperture needed to create this effect, so it cannot be said to be a case of optical isolation under the definition of this paper. If the concept of multiple image formation is extended to include image separation in an axial dimension, we find examples of nonlocalied optical effects creating multiple images in contact lenses and intraocular lenses that employ diffractive optics. 10 They are forms of Fresnel lenses in which the phase change at transitions is designed to be a half-wave instead of a full wave. In this sense they are quite like the phase gratings alluded to above. * The author is Chief Technologist at Humphrey Systems. REFERENCES 1. Lord Rayleigh, Philos. Mag. 8, 261 (1879). 2. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), p J. W. Goodman, Introduction to Fourier Optics (McGraw- Hill, San Francisco, Calif., 1968), p J. Y. Wang and D. E. Silva, Wave-front interpretation with Zernike polynomials, Appl. Opt. 19, (1980). 5. D. Malacara, Optical Shop Testing (Wiley, New York, 1978), pp , J. Schwiegerling and J. E. Greivenkamp, Using corneal height maps and polynomial decomposition to determine corneal aberrations, Optom. Vision Sci. 74, (1997). 7. H. Howland and B. Howland, A subjective method for measurement of monochromatic aberrations of the eye, J. Opt. Soc. Am. 67, (1977). 8. C. Campbell, Corneal aberrations, monocular diplopia and ghost images: analysis using corneal topographical data, Optom. Vision Sci. 75, (1998). 9. C. Campbell, Caustics in astigmatic systems, in Vision Science and Its Applications, Vol. 1 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp S. A. Klein and Z. Ho, Multione bifocal contact lens design, in Current Developments in Optical Engineering and Diffraction Phenomena, R. E. Fischer, J. E. Harvey, W. J. Smith, eds., Proc. SPIE 679, (1986).