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`Fundamentals of macro axial gradient index
`optical design and engineering
`
`Paul K. Manhart, MEMBER SPIE
`Richard Blankenbecler, MEMBER SPIE
`Cheshire Optics
`974 Cottrell Way
`Stanford, California 94305
`E-mail: pmanhart@worldnet.att.net,
`rblankenbecler@worldnet.att.net
`
`Abstract. Homogeneous lens material is characterized by an index of
`refraction and a point on the glass map nd⫽f(␷d). Gradient refractive
`index (GRIN) lenses have a spatially varying index and dispersion and
`are represented by a line on the glass map. GRIN lenses open the door
`to a wide variety of optical design applications incorporating entire lenses
`of axial gradient refractive material (macro-AGRIN). Axial gradient mate-
`rial essentially gives biaspheric behavior to lenses with spherical sur-
`faces and exhibits a controlled gradient in both index and dispersion.
`Thus, the applications for this material range from simple singlet lens
`used for imaging laser light, in which spherical aberration is eliminated,
`to complex multielement lens systems, where improved overall perfor-
`mance is desired. The fusion/diffusion process that produces this mate-
`rial is surprisingly simple, repeatable, and applicable to mass production.
`The advantages of AGRIN technology coupled with the recent advances
`in material development and its accessibility in commercially available
`lens design programs provides optical designers with the opportunity to
`push the performance of optical systems farther than with conventional
`optics.
`© 1997 Society
`of Photo-Optical
`Instrumentation Engineers.
`[S0091-3286(97)00406-6]
`
`Subject terms: optical materials; optical systems; gradient index; axial gradient
`refractive index; gradient refractive index; radial gradient refractive index; macro
`axial gradient, radial gradient; optical design; glass lines; unique optical systems;
`high performance; asphere.
`
`Paper MAT-04 received Aug. 1, 1996; revised manuscript received Nov. 21,
`1996; accepted for publication Dec. 16, 1996.
`
`1 Introduction
`Traditional optical systems are based on the use of homo-
`geneous refractive materials with spherical surfaces. A
`spherical surfaced lens will produce images with intrinsic
`optical aberrations. Basic optical principles lead to the use
`of multiple elements in a lens system to provide imagery
`across a required field with acceptably low aberration con-
`tent. Fabrication of spherical surfaces can be carried out
`using several traditional methods to extremely high accu-
`racy and smoothness.
`Improved images with reduced aberrations can be ob-
`tained by using aspheric surfaces. The production and test-
`ing of nonspherical surfaces is more complex and consid-
`erably more
`expensive
`than for
`spherical
`surfaces.
`Generally, aspheric surfaces will cost about 10 times that of
`a spherical surface. Nevertheless, cost efficient mass pro-
`duction of optical elements with aspheric surfaces can be
`accomplished with plastics or glasses, usually to acceptable
`accuracy for some applications. While mass production of
`molded aspherics can be cost competitive, it is subject to
`extremely high startup costs, limited material selection and
`small diameters.
`Axial gradient refractive index 共AGRIN兲 offers an ap-
`proach that would build specific aspheric effects into the
`refractive properties of the material while permitting the
`use of traditional spherical surface finishing methods. The
`refractive power at each point on a surface is determined by
`
`Snell’s law, given in Eq. 共1兲, where ␪ is the angle of inci-
`dence the ray makes
`
`n sin ␪⫽n⬘ sin ␪⬘
`
`共1兲
`
`with the surface normal, ␪⬘ is the angle after refraction, n is
`the index of refraction of the incident medium and n⬘ is the
`index of the refractive medium. The shape of the surface
`determines the progressive change in angle of incidence
`across the aperture that leads to image formation. This
`same relation determines the intrinsic aberration that will
`be produced by the surface. Additional control of the ray
`passage through a surface can be gained by using a material
`in which the index of refraction varies with the location on
`the surface and/or within the medium. If the refractive in-
`dex varies along the optical axis, it is known as an axial
`gradient. If the index varies with radius, it is known as a
`radial gradient.
`
`2 Axial Gradients
`The term macro-AGRIN refers to a solid piece of axial
`gradient optical material with relatively large diameter,
`thickness and change in index (⌬n兲, and is used inter-
`changeably with the term AGRIN throughout this paper.
`
`Opt. Eng. 36(6) 1607–1621 (June 1997)
`
`0091-3286/97/$10.00
`
`© 1997 Society of Photo-Optical Instrumentation Engineers
`
`1607
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`
`Manhart and Blankenbecler: Fundamentals of macro axial gradient index . . .
`
`Macro-AGRIN is made via a high temperature fusion/
`diffusion process described later in this text that turns a
`series of discrete glasses on the glass map into glass lines
`共see Fig. 1兲. An axial gradient is typically defined by a
`polynomial function in z,
`the distance along the optical
`axis. A common expression for this type of gradient is
`
`n 0⫽n 00⫹n 01z⫹n 02z 2⫹n 03z 3⫹ . . .
`
`.
`
`共2兲
`
`The most significant way an axial gradient affects aberra-
`tions is by the variation of index over the curved surface of
`the lens. Snell’s law states that there are only two factors
`that determine how light is bent when it hits a glass surface.
`The first factor is the angle of incidence ␪of the ray on the
`surface, and the second factor is the index of refraction
`n⬘ at the point of contact. An aspheric surface controls
`aberrations by changing the localized radius of curvature
`共surface normal兲 on the lens surface, thus changing the
`angle of incidence for a particular ray. An axial gradient
`lens affects aberrations by leaving the surface spherical and
`changing the index of refraction through the material.
`When a curved surface is put on a plane parallel piece of
`this material, the gradient is exposed because the curve of
`the surface exposes different depths z in the material. This
`results in a changing index value tracking radially outward
`along the curved surface. A lens made from a solid block of
`axial gradient material has two surfaces exposing the gra-
`dient 共front and back兲, thereby giving each lens the equiva-
`lent of two aspheric surfaces. The contribution to aberration
`at the surface of a lens, due to a gradient material is re-
`ferred to as a surface term. In addition to the two surface
`terms, there is a transfer term as the wavefront propagates
`through the lens, between the two surfaces. In this region,
`the light travels in a curve as the wavefront encounters a
`continuous change in index and dispersion, identified by
`the Abbe number,
`
`␯d⫽
`
`n d⫺1
`n F⫺n C
`
`.
`
`共3兲
`
`It is important to think in terms of wavefronts, as opposed
`to rays, when understanding how light travels through a
`gradient material. A wavefront always bends toward the
`high index, where light travels slower, similar to a march-
`ing band turning a corner.
`Adding up the contributions from the two surface terms
`and the transfer term can give AGRIN an advantage over
`aspheric lenses. In addition, the spherical surfaces of lenses
`made of AGRIN are self generating, easy to test, plate fit to
`high accuracy and no more expensive to fabricate than con-
`ventional surfaces. The effect on aberrations from a macro-
`AGRIN lens can be adjusted by changing the lens shape
`factor or gradient profile1 关Eq. 共2兲兴. A multielement lens
`system can benefit from a gradient index lens by relaxing
`the requirements of the other lenses. The gradient material
`essentially acts like a ‘‘work sponge,’’ soaking up a portion
`of the work that would have previously been distributed
`among all the elements. Designing functionality into an op-
`tical material helps relax optical systems and paves the way
`to improved performance. Improved performance can be in
`the guise of smaller, cheaper, sharper, lighter or stronger
`products.
`
`1608 Optical Engineering, Vol. 36 No. 6, June 1997
`
`The importance of gradient index materials have been
`realized for over 40 years.2–4 Extensive theoretical explo-
`ration and conceptual designs can be found in many scien-
`tific articles.5–8 The poor reputation of GRIN material,
`however, comes from stagnation in the material develop-
`ment. Small size 共up to 3 mm兲, poor repeatability, and high
`cost are associated with GRIN materials. Four major differ-
`ent processes, i.e., ion exchange, sol-gel, chemical vapor
`deposition 共CVD兲, and high temperature fusion/diffusion
`are currently used to produce GRIN materials. Recogniz-
`able commercial products are Selfoc™ produced by Nip-
`pon Sheet Glass via an ion-exchange process, graded-index
`optical fibers via the CVD process and axial gradient laser
`singlets via fusion/diffusion. Theoretically, radial GRIN is
`more versatile than AGRIN because the radial refractive
`index gradient directly contributes to the optical power,
`Petzval and spherical aberration of the optical system.
`However, by the same token, this makes the performance
`of the radial GRIN lens more sensitive to manufacturing
`variations and repeatability in the gradient index profile
`than AGRIN lenses. A radial gradient lens will directly
`affect the first and higher order properties of an optical
`system because the gradient is perpendicular to the wave-
`front propagation. Repeatability is an important issue with
`radial GRIN.
`The contribution to aberration correction from the trans-
`fer term of a thin axial gradient lens is secondary in that the
`gradient does not directly contribute to the optical power or
`aberrations of the lens, but rather, it is the exposure of the
`gradient along a curved surface that is responsible for most
`of the aberration correction. The gradient in this case is
`essentially parallel to the wavefront propagation, so the
`transfer term is not as dominant as it is with radial gradient.
`Therefore, the AGRIN lens performance is more tolerant to
`small manufacturing variations in the profile compared to
`the radial GRIN lens.
`
`3 Fusion/Diffusion Process (Macro-AGRIN)
`A process to produce solid blocks of AGRIN is based on
`controlled fusion/diffusion of different glasses at a high
`temperature. The process, developed by Hagerty and
`Pulsifer,9 starts by identifying a specific family of glass
`compositions that span the desired range in optical index
`and dispersion. AGRIN glass is created by fusing together a
`stack of discrete molten glass layers, where each constitu-
`ent layer has a distinctive composition and desired optical
`property. After a controlled diffusion process at high tem-
`perature 共see Fig. 2兲, the glass layers fuse and diffuse into
`one piece of gradient index glass with a smooth variation of
`optical properties throughout. Prescribable index profiles
`共linear, quadratic and cubic兲, in lead and crown glass com-
`positions have been achieved in large diameters, thickness
`and index variation.10–13 The initial step function index
`profile of the layers is eliminated via the diffusion of con-
`stituent atoms within and across the layer boundaries. This
`process results in a smooth variation of index. By varying
`the index and thickness of each layer, and by controlling
`diffusion temperature and time, AGRIN can be produced
`for arbitrary monotonic index profiles. The profiles are
`typically monotonic because of the separation of the con-
`stituent molten glasses based on density. High temperature
`diffusion 共1100 °C兲 yields high diffusion rates for all com-
`
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`Manhart and Blankenbecler: Fundamentals of macro axial gradient index . . .
`
`Fig. 1 Glass map showing refractive index as a function of Abbe number.
`
`ponents in the glass, therefore, a wide variety of glass fami-
`lies can be used to produce AGRIN with different index/
`dispersion relationships 关n d⫽ f (␷d)兴. Profiles with steep
`slopes (dn/dz), experience more stress than profiles with
`small slopes. This stress is due to mismatches in the coef-
`ficient of thermal expansion of the constituent glasses after
`diffusion. The resultant stress manifests itself as residual
`asymmetric radial terms in the AGRIN material, limiting
`the useful aperture over which an optical quality wavefront
`can be obtained. Steeper sloped gradients have smaller use-
`ful diameter than shallower sloped gradients because of the
`residual stress associated with a more rapid change in ma-
`terial composition.
`Two key factors of the fusion/diffusion process are 共1兲
`the selection of base glass compositions and 共2兲 the diffu-
`sion process. The glass compositions must be compatible
`with each other, so that the compositional changes due to
`the diffusion between glass layers will not cause undesir-
`able property changes, such as thermal expansion, phase
`separation and devitrification. The step function-like index
`profile between layers is eliminated via the diffusion of
`constituent atoms within and between the layers during the
`process; this eventually results in smooth variation in index
`
`as well as other physical and material properties. The pro-
`cess is illustrated in Fig. 2.
`
`4 Diffusion Software
`To prescribe the index profile, it is necessary to 共1兲 estab-
`lish the relation between glass properties, such as refractive
`index, dispersion,
`thermal expansion coefficients, glass
`transition temperature, and glass composition, and 共2兲 un-
`derstand and simulate the diffusion process. The design
`problem for prescribing the refractive index profile is to
`find the desirable glass family and an achievable initial in-
`dex distribution n(z,0) that will yield the desired index
`profile n(z,t) to the accuracy required after diffusion at
`temperature T for a time t. In the fabrication methods al-
`ready described, the index of each layer can be fixed for a
`given sample batch; the design variables are then the thick-
`ness of each layer, the diffusion temperature T and time
`t. Diffusion simulation software14,15 determine these pro-
`cess variables quickly. The diffusion model is based on
`multicomponent interdiffusion theory. All constituents of
`
`Optical Engineering, Vol. 36 No. 6, June 1997
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`Manhart and Blankenbecler: Fundamentals of macro axial gradient index . . .
`
`Fig. 2 Fusion/diffusion process for axial gradient material fabrication.
`
`the glass participate in diffusion at high temperature. The
`diffusion constants of these constituents are also functions
`of their elemental concentrations.
`The change in the index distribution during the diffusion
`process was simulated using the diffusion software and is
`illustrated in Fig. 3 for a seven-layer sample. It shows that
`diffusion at the high index region is much faster than at the
`low index side. This is expected because of the higher lead
`densities in the higher index material. After being diffused
`at high temperature for the required time, the index profile
`becomes smooth and has a linear index gradient region be-
`tween 1.580 and 1.750. Using the parameters established
`by the diffusion software, the calculated index profiles are
`
`in excellent agreement with the produced index profiles,
`demonstrating the effectiveness of the software in prescrib-
`ing the index profile for specific optical designs.10–12 Figure
`4 plots the calculated index profile along with the measured
`data of the produced profile. The results from different dif-
`fusion experiments also show excellent repeatability in
`both the index profile and optical quality.
`The unique aspect of this process for producing macro
`gradient index glass is that layers of glass possessing dif-
`ferent physical properties can be fused together to produce
`a repeatable predetermined index gradient. The controlled
`diffusion of glass components such as Pb, Ba and La, pro-
`vides a gradual transition in properties from layer to layer.
`
`Fig. 3 Diffusion simulation for seven-layer profile as a function of diffusion time t at 1000 °C.
`
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`Manhart and Blankenbecler: Fundamentals of macro axial gradient index . . .
`
`Fig. 4 Comparison of the produced index profile with the profile calculated from diffusion software.
`
`Carefully chosen glasses enable this high temperature dif-
`fusion process to produce gradient
`index glass having
`macro size, high ⌬n and good wavefront quality.
`
`5 Optical Design
`Macro-AGRIN offers optical designers with a unique op-
`portunity to incorporate lenses made from solid blocks of
`gradient material into their optical systems. Gradient mate-
`rials are described by profiles. For axial gradients, the pro-
`file is a plot of the index of refraction at a reference wave-
`length 共typically 0.58756 ␮m兲, as a function of axial
`position within the blank 关Eq. 共2兲兴. Profiles can be for either
`increasing or decreasing index as a function of position.
`The slope is negative for a decreasing index and positive
`for an increasing index. Some fundamental insights into the
`behavior of lenses made from this material follow, along
`with a process to define and model the dispersive properties
`of the gradient.
`
`5.1 LinearVersusNonlinearProfile
`New issues have emerged while designing optical systems
`with macro-AGRIN materials. For instance, it has been
`found that nonlinear profiles are more advantageous than
`linear profiles because the relative slope of the gradient
`over each lens surface can be different.1 Figure 5共a兲 shows
`calculated third order spherical aberration, 共SA3兲 of an
`F/3 singlet lens, plotted as a function of lens shape factor
`X, for a linear gradient range of ⫺0.4⭐⌬n⭐0.4. The dif-
`ferent curves represent different slopes, or ⌬n’s. For a lens
`with the object at infinity, minimum SA3 occurs near a
`shape factor, or bending, of ⫹1.0, which corresponds to a
`convex first surface and a flat rear surface. To correct an
`
`F/3 singlet for SA3 using a linear gradient, ⌬n must be
`between ⫺0.03 and ⫹0.25. Once that condition is met,
`there exists a solution for zero SA3. The most prominent
`aspect of this plot however, is that the curves tend to pivot
`about a point that has a shape factor of zero 共equiconvex/
`convex兲. The phenomena that we call the pivot point has
`been observed for all cases analyzed, including root mean
`square 共rms兲 spot size, coma, astigmatism and distortion.
`The pivot point also exists for spherical type gradient ge-
`ometry’s 共but shifts as a function of the exact radius of the
`isoindicial surfaces兲. In essence, the existence of the pivot
`point implies that a monotonic linear profile throughout the
`material has no effect in controlling SA3 when the shape
`factor of the lens is near zero. Wang hints at this in his
`dissertation.16 For an equiconvex lens with a linear axial
`gradient, the net effect of the gradient on spherical aberra-
`tion is zero because the two spherical surface curvatures, of
`equal and opposite sign, have the same gradient slope over
`the saggita z 共sag兲, given by
`
`z⫽
`
`cr 2
`1⫹ 共1⫺c 2r 2兲1/2 .
`
`共4兲
`
`The contribution to spherical aberration from one gradient
`surface is canceled by the other gradient surface. For a lens
`bending ⫺1⭐X⭐⫹1, a linear axial gradient lens’ optical
`performance is compromised by one of the surfaces. Me-
`niscus lenses, however, have the property that the gradient
`contribution to spherical aberration is additive on both sur-
`faces because their curvatures are of the same sign.
`The existence of the transfer term, and the fact that the
`marginal ray heights are different on each surface for a
`
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`Manhart and Blankenbecler: Fundamentals of macro axial gradient index . . .
`
`Fig. 5 Third order spherical aberration versus lens shape factor for (a) a linear gradient profile and (b)
`for a quadratic gradient profile.
`
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`Manhart and Blankenbecler: Fundamentals of macro axial gradient index . . .
`
`dient slope over the two surfaces of a lens. The first, Fig.
`6共a兲, shows two separate slopes, of opposite sign, over each
`surface. In this case, the gradient surface terms are additive
`for shape factors between ⫾1 and subtractive for meniscus
`shapes. Processes such as ion exchange will produce such
`profiles, of equal but opposite slope, but the small penetra-
`tion depth of gradient into the material limits the useful
`diameter of the optics. Figure 6共c兲 shows a grin cap,10
`which essentially puts a gradient over one side of the lens
`and makes the other side homogeneous. In the case of a
`grin cap lens there is only one surface term. To make a grin
`cap lens, two lenses would be made 共one gradient and one
`homogeneous兲 and cemented together. This adds complex-
`ity to the fabrication process. Figure 6共b兲 shows a natural
`quadratic profile that starts out shallow on the first surface
`and ends up steep on the second surface. The fusion/
`diffusion process that makes macro gradient naturally
`comes out in a sigmoid profile, flat on the top, steepest in
`the center, and flat at the end. Careful positioning of the
`lens within this type of profile allows for many more solu-
`tions to SA3 than with a linear gradient profile.
`
`5.2 AsphereVersusAGRIN
`To correct for a given amount of spherical aberration, the
`steepness of a gradient slope scales inversely as a function
`of lens aperture. The smaller the lens, the steeper the slope
`required to do the same amount of work as a larger lens.
`For example, the amount of ⌬n required to correct the
`spherical aberration of a 10 mm diameter f /2 singlet is the
`same as that required for a 50 mm diameter f /2 singlet.
`Since ⌬n is a constant, but ⌬z, the difference in sag be-
`tween the two lenses is not, the slope ⌬n/⌬z scales in-
`versely with aperture. Some of the issues relating the lens
`design process with the material fabrication process include
`practical limits to lens diameter and the quality of the trans-
`mitted wavefront. From a manufacturing perspective,
`steeper sloped materials are currently better suited to
`smaller aperture optics and shallow sloped materials may
`be used for larger diameter optics. The reason for this, is
`that residual thermal stress in the material is a function of
`⌬n and slope. Steeper sloped materials have higher stress
`which results in stronger residual radial terms distorting the
`transmitted wavefront in an asymmetric way. The higher
`stress results from the material changing rapidly over a
`short distance . To maintain comparable wavefront quality
`to that of shallower sloped material, the aperture must be
`reduced. Since this is a material and manufacturing issue,
`and not a fundamental
`limitation,
`these issues may be
`worked out in time.
`To compare a lens with an aspheric surface to a macro-
`AGRIN lens requires some definitions. A direct comparison
`of AGRIN material to an asphere is difficult because of the
`transfer term associated with gradient material. To isolate
`the surface term of an AGRIN lens, it is necessary to ex-
`amine a plano-convex lens, with the plano side facing a
`collimated wavefront. In this special case, the transfer term
`is zero and the plane wave propagates through the axial
`gradient undeviated. The wavefront is only affected by the
`gradient on refraction at the second surface, where the
`variation of index over the curved surface affects the bend-
`ing of light differently than a constant index would. A
`
`Optical Engineering, Vol. 36 No. 6, June 1997
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`
`Fig. 6 Three methods of altering gradient slope over each curved
`surface of a lens: (a) two cemented linear gradients, (b) nonlinear
`profile, and (c) grin cap.
`
`given ray, shifts the pivot point slightly away from the zero
`shape factor. The proximity of the pivot point to the zero
`shape factor is direct evidence that the inhomogeneous
`transfer term is small compared to the surface term, and
`thus not a significant variable for monochromatic designs
`of thin AGRIN lenses. One consequence of the pivot point
`is that SA3 can be made to change sign with bending if
`⌬n is large enough. Another, is that less ⌬n is required for
`positive bendings than for negative bendings to achieve this
`sign reversal. This is a direct consequence of the minimum
`SA3 occurring for a shape factor near ⫹1.0 for a homoge-
`neous lens. The pivot point provides useful insight when
`designing optical systems with this material. To maximize
`the potential of macro-AGRIN, lens shape factors near zero
`should be avoided, and lenses with shape factors between
`⫾1 should be viewed with suspicion. There are situations
`however, in which it is desirable to have a lens with a shape
`factor near zero. Such situations might exist when trying to
`incorporate gradients into existing designs with minimal
`amount of redesign, when trying to balance other aberra-
`tions, or when working with finite conjugates. To effec-
`tively apply axial gradients to near equiconvex or concave
`lenses the gradient contribution over each surface must be
`changed by adjusting the relative slope of the gradient over
`each of the lenses curved surface.
`Nonlinear sigmoid type gradient profiles can help ad-
`dress the problem of the pivot point cancellation. The slope
`of a nonlinear gradient may start out high and end low or
`visa versa. The nature of a nonlinear gradient therefore al-
`lows the GRIN aberration contribution on each surface to
`be changed by altering the relative slope of the gradient
`over each surface. Thus, nonlinear axial gradients are more
`effective than linear axial gradients because there is less
`compromise for lens bendings between ⫾1. Figure 5共b兲
`shows the same plot as Fig. 5共a兲 for a nonlinear, monotonic
`共quadratic兲, gradient profile. There are many more solutions
`for zero SA3 using nonlinear gradients than for linear gra-
`dients. Note in Fig. 5共b兲 that SA3 does not change sign as it
`does for the linear gradients in Fig. 5共a兲. Once a particular
`⌬n hits zero SA3, it stays there for the remaining shape
`factors. Although the pivot point does not change with non-
`linear gradients, there are many more useful solutions at
`other shape factors.
`Figure 6 shows several ways to change the relative gra-
`
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`Manhart and Blankenbecler: Fundamentals of macro axial gradient index . . .
`
`Fig. 7 Delta index required to correct spherical aberration as a function of lens f-number.
`
`plano-convex lens such as this, with a shape factor
`X⫽⫺1 has a fair amount of negative spherical aberration.
`The shape factor for minimum SA3 of a homogeneous lens
`is near a convex-plano lens (X⫽⫹1), where the convex
`side faces the infinite conjugate. The amount of gradient
`and/or asphericity required to correct a plano-convex lens
`for SA3 will be more than it would for a convex-plano, but
`the relative difference between ⌬n and maximum aspheric-
`ity reflect the work that the isolated surface terms are con-
`
`tributing to spherical aberration. Figure 7 shows a plot of
`⌬n required to correct spherical aberration as a function of
`lens f -number for surface contribution only. The ⌬n values
`in these plots show what is required to correct all orders of
`spherical aberration for a one inch diameter, f /2 singlet of
`shape factor X⫽⫾1.
`Figure 8 shows the same plot as Fig. 7, for an aspheric
`lens of shape factor X⫽⫾1. The y axis represents the
`
`Fig. 8 Normalized aspheric sag required to correct spherical aberration as a function of
`f-number.
`
`lens
`
`1614 Optical Engineering, Vol. 36 No. 6, June 1997
`
`8
`
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`
`Manhart and Blankenbecler: Fundamentals of macro axial gradient index . . .
`
`maximum amount of aspheric sag deviation required to cor-
`rect spherical aberration. Since the maximum aspheric sag
`of a surface scales with radius of curvature, the aspheric
`sag values plotted in Fig. 8 are normalized to the radius of
`curvature of the aspheric surface.
`
`6 Dispersion Modeling: A Walk Along the Glass
`Line
`All points within homogeneous elements have the same
`value of index and are the same function of wavelength.
`GRIN elements have a spatially varying index function.
`The GRIN transfer term depends on the spatial derivative
`of the index. The index and its spatial derivative have dif-
`ferent dispersion properties. This difference allows chro-
`matic corrections to be dealt with in new ways in systems
`with GRIN elements.
`Commercially available optical design software has
`been able to trace rays through an inhomogeneous material
`for many years. Some of the more popular profiles and
`geometries are programmed for user convenience. When a
`gradient geometry is selected 共such as axial, radial, or
`spherical兲, the user may define the coefficients to describe a
`specific index variation 共profile兲, throughout the material.
`These coefficients may also be variables during optimiza-
`tion.
`It is generally known that index of refraction for optical
`materials is a function of wavelength.17 This is also true for
`macro-AGRIN material where the index n is a function of
`Abbe number ␷ 关Eq. 共3兲, Fig. 1兴. The index profile is typi-
`cally described by a polynomial 关Eq. 共2兲兴. Because of dis-
`persion, the coefficients describing this profile vary accord-
`ing to wavelength. Designing white light optical systems
`with gradients requires the designer to control the relation-
`ship between ⌬n and ⌬␷if the resultant gradient material is
`to be realistic. Since AGRIN is represented by glass lines,
`or monotonic curves, on the glass map, this relationship can
`be predefined for each specific family of glasses and thus
`transparent to the optical designer. Each constituent glass
`comprising the family has its own unique set of Buchdahl
`coefficients describing its dispersion. Fitting these Buch-
`dahl coefficients to a function in index through the full
`range of the material links the dispersion coefficients to
`material position within the material.
`Given the Buchdahl dispersion expression as follows,
`
`␦n⫽V 1␻⫹V 2␻2⫹V 3␻3⫹V 4␻4,
`
`where
`
`␦n⫽n i⫺n d , ␻⫽
`
`␦␭
`1⫹␤共␦␭ 兲 , ␤⫽
`
`1.0
`␭ d⫺0.187
`
`,
`
`␦␭⫽␭ i⫺␭ d ,
`
`␭ d⫽0.58756 ␮m.
`
`Each glass in the family has its own set of coefficients,
`V 1 , V 2 , V 3 , and V 4 , to describe its dispersion. To model
`the gradual variation in chromatic properties within the gra-
`dient material after each glass has been fit to Buchdahl, the
`coefficients V i are fit to a cubic function in n d , defined as
`the index at the reference wavelength 共typically 587.56
`nm兲. The result looks like
`
`
`
`2兲⫹␥i共n d兲⫹␦i .V i⫽␣i共n d3兲⫹␤i共n d
`
`
`
`共5兲
`
`Since the Buchdahl fit in this case is carried out to the
`fourth term, there are four values of alpha, beta and gamma
`and delta each. Once n d(z) is known from the reference
`profile through the material 关Eq. 共2兲兴, the Buchdahl coeffi-
`cients can be calculated at that location. Once the Buchdahl
`coefficients are known, so is the index at any wavelength.
`This methodology has been implemented in several of the
`commercially available software codes where each AGRIN
`glass line has its own unique name that the designer can
`call out. Once the type of AGRIN glass is chosen 共flint,
`crown etc.兲, the designer only needs to be concerned with
`the index profile at 587.56 nm wavelength. The software
`will fill in the indices at all the system wavelengths. This
`model contributes only 1⫻10⫺5 uncertainty17 in index and
`may be implemented from catalog data of constituent
`glasses prior to diffusion, or measured index data at several
`wavelengths after diffusion.17
`For lens designers who want to use off-the-shelf AGRIN
`profiles, a gradient surface type has been developed allow-
`ing the use of predefined profiles that can be called by
`name. This capability can be found in CODE V 共CODE V
`Optical Design Software is a product of Optical Research
`Associates, Pasadena, California兲, OSLO 共OS