WO1997018170A1

WO1997018170A1 - Refractive elements with radially graded properties and methods of making same

Info

Publication number: WO1997018170A1
Application number: PCT/US1996/018623
Authority: WO
Inventors: Richard Blankenbecler
Original assignee: Richard Blankenbecler
Priority date: 1995-11-16
Filing date: 1996-11-15
Publication date: 1997-05-22

Abstract

Methods for making refractive optical elements with prescribed gradient physical properties that possess cylindrical symmetry are taught. These physical properties include, but are not restricted to, the optical index of refraction. The final spatial distribution of constituents is calculated in assemblages with different starting parameters. The initial assemblage that results in the prescribed final distribution is chosen for fabrication. The assemblage is then rotated to control the gravitational mobility of the atomic constituents while being heated and subjected to suitable diffusion conditions. The starting assemblage and diffusion regimen can thus be selected to yield a final desired property distribution. These are then used in the fabrication of an optical element with the desired radial gradient properties.

Description

REFRACTIVE ELEMENTS WITH RADIALLY GRADED PROPERTIES AND METHODS OF MAKING SAME

Field of the Invention:

The present invention relates generally to refractive optical components that form preforms, lenses and fibers having a radial gradient property and more particularly to optical components formed subject to rotating fields to have a spatially varying radial gradient of an active atomic species.

Description of the Related Art:

The most common optical components in present use, often also generically referred to as elements, are formed from homogeneous transparent materials. Compound elements can be formed by joining together homogeneous sub-elements with the resulting element possessing piecewise continuous physical properties. Optical elements that have a physical property that varies internally are commonly referred to as "gradient property" elements. Optical designers have long realized the advantages that could be obtained by use of optical elements with continuous radial gradient properties, such as index of refraction. These designers have also recognized that any practical use would also require a practical method of closely if not precisely prescribing the final shape or profile of a radial gradient element prior to fabrication of the element. The techniques available for the manufacture of gradient property elements are limited. The gradient properties may be axial, i.e. , varying along the axis of the element, or radial, / e , varying along the perpendicular to an axis. Elements with axial gradient profiles can be fabricated by ion exchange, chemical vapor deposition and the SOL-GEL process Ion exchange and chemical vapor deposition can also be used to fabricate small elements with radial symmetry. These techniques, particularly as applied to the fabrication of radial symmetry elements, can produce only small changes in physical properties, such as index of refraction, over a limited region. Further, and most importantly, the profile of the chosen property cannot be accurately or often even generally prescribed when using these processes. Macro-gradient elements with prescπbable axial profiles and with larger index changes can be fabricated by a diffusion process as is described in U.S. Pat. Nos. 4,907,864, issued 1 1 /28/89; 4,883,522 issued 03/1 3/90; and 4,929,065 issued 05/29/90 all by Hagerty, et al. These processes are particularly limited to the fabrication of axial gradients. Axial gradient elements produced by these processes can be used as lens blanks and their surfaces subsequently ground and polished to function as lenses. Alternatively, the axial gradient lens blanks can be slumped over molds to form spherical, pseudo- cylindrical, or more general index profiles as is described by the author and M. Wickson in U.S. Pat. No. 5,236,486, issued 08/17/93. Slumping is a well known process involving the heating of a prefabricated initially planar lens blank while the blank rests on a mold until the blank deforms to the surface shape of the mold.

None of the above methods can fabricate radial gradient macroelements with a cylindrical symmetry. This is an element whose profile varies radially out from the optical or z-axis and is rotationally symmetric around this axis. A macro-element is defined here to be a cylindrical element with a diameter larger than roughly one-half centimeter. One such lens element with cylindπcal symmetry and a changing index of refraction is described by W. Ewald in U.S. Pat. No. 1 ,943,521 , issued 01 /1 6/34. This element is built up of homogeneous constituent subelements. The assembly is not then diffused, but is cemented or otherwise bonded together. The assemblage is then subsequently ground into a conventional lens shape. The piecewise staircase index variation, which is directly defined by the step concentration variation of the subelements, can be chosen to minimize spherical aberrations. However, this lens construction process is capable only of reducing and not eliminating spherical aberrations. Even without involving a diffusion process step, such lenses are difficult to fabricate to the accuracy required for contemporary good optical performance. In U.S. Patent No. 5,262,896, "Refractive Elements with Graded

Properties and Methods of Making Same," issued to Blankenbecler on 1 1 /16/93, a method for producing prescribable axial gradient profiles is taught. This method is only applicable to stationary diffusion processes as specifically applied to planar materials. That is, the planar diffusion processes contemplated do not include any effects of motion or in change in the applied gravitational field. Specifically, the methods provide for choosing an initial stack of homogeneous planar layers of glass, each with a potentially differing index and thicknesses, so that after a diffusion procedure, a prescribed axial index profile is achieved. A general description of atomic diffusion processes is found in "Diffusion in Solids" by P.G. Shewmon, McGraw-Hill, 1 963.

Consequently, there is a need for a method of choosing the parameters of an initial assemblage of glass materials so that a cylindrical element can be fabricated with a final radial gradient profile, such as index of refraction, while fully maintaining cylindrical symmetry.

Summary of the Invention Thus, a general purpose of the present invention is to provide a method of selecting an initial assemblage of cylindrical tubes and core, the materials of the assemblage, and the rotation and diffusion process parameters appropriate achieve a final radially symmetric spatial distribution of an active atomic species to realize a corresponding radial or cylindrical symmetric physical characteristic. This is achieved in the present invention through a process of choosing an initial assemblage of materials each of which has a given concentration of an active atomic species, determining a set of process parameter including time, temperature and rotation rate applied to the assemblage that yield a minimum deviation from a desired final radially symmetric spatial distribution of the active atomic species within the assemblage, and using the chosen initial assemblage and set of process parameters to form a final preform structure having the final radially symmetric spatial distribution of the active atomic species. In determining the set of process parameters, a rotation rate is determined based on the materials of the assemblage and the temperature applied during rotation of the assemblage to facilitate controlled interdiffusion of the materials of the assemblage such that the applied gravitational field and atomic motility of the active atomic species within the assemblage average to zero. The determination is further based on the physical constraint upon diffusion established by an encapsulant barrier at the outer surface of the assemblage during interdiffusion of the materials of the assemblage.

Physical characteristics that can be controlled so as to have a prescribed spatial distribution include, but are not limited to, the index of refraction, coefficient of thermal expansion or CTE, and thermal conductivity. The example of the index of refraction will be used herein where specificity improves clarity.

Thus, an advantage of the present invention is that it provides an improvement in the Ward-Pulsifer process. This improvement allows a fabricator to choose an initial assemblage and process rotation rate, diffusion time, and temperature that will produce a prescribed gradient profile in the finished symmetrical radial gradient cylindrical element. Another advantage of the present invention is that it provides a method of allowing the fabricator to choose an optimal initial assemblage of glass types that will result in the prescribed gradient profile. The method also allows the fabricator to weigh different aspects of the optical system to achieve desired ends, among which are minimizing cost, improving performance, minimizing weight, simplifying assembly, and enhancing reproducibility by increasing the margin of stability.

Still another advantage of the present invention is that it provides a general method of allowing the fabricator to determine if a lens profile can be practically fabricated without resort to laboratory trial and error. Costly situations in which a fabricator produces a complete optical system design containing a radial gradient element that cannot be theoretically or practically fabricated can be avoided.

Yet another advantage of the present invention is that it provides the fabricator with a method of determining accurate corrections to the time, temperature, and rotation rate parameters of the fabrication process to still obtain a prescribed gradient profile in the finished symmetrical radial gradient element Particularly in mass-production circumstances, the method of the present invention allows for the controlled correction of process parameters to take into account the batch-to-batch or lot-to-lot property variations of the starting optical element materials.

Brief Description of the Drawings

These and other advantages and features of the present invention will become better understood upon consideration of the following detailed description of the invention when considered in connection of the accompanying drawings, in which like reference numerals designate like parts throughout the figures thereof, and wherein: Figure 1 a illustrates the geometry of an initial assemblage of fitted cylindrical tubes and core;

Figure 1 b illustrates an end-on view of assemblage with four cylinders and the coordinate system used in the mathematical description of the present invention;

Figure 2 provides graphs of concentration versus radius for a monotonic density assembly showing the initial, intermediate and final concentration profiles of a diffusing element for the assemblage;

Figure 3 provides graphs of concentration versus radius for a varying density assembly showing the initial, intermediate and final concentration profiles of a diffusing element for the assemblage; and

Figure 4 provide schematic flow diagrams of the process design logic used in a computer program for carrying out a preferred embodiment of the present invention.

Detailed Description of the Invention

The present invention provides for improved methods for controlling the fabrication of gradient optical elements having a required radial distribution of one or more physical properties. In the preferred embodiment, the initial optical element is constructed from one or more glass tubes provided concentrically around a glass core. The structure and composition of the tubes and core and the manner of initially forming an exemplary element upon which the present invention may be used is described in detail in the copending application entitled "Glass Preform with Deep Radial Gradient Layer and Method of Manufacturing Same" (S/N 08/025,079; hereinafter "Ward-Pulsifer"). In brief, an optical preform is constructed from an initial cylindrical rod or core inserted through a matching cylindrical hollow in a tube. Multiple tubes may be used.

The material of the core and tubes may be chosen from any of a number of families of optical materials, typically referred to as glassy materials, including lead silica, borosilica, and other glass families that are well known in the art. Typically, each family of glasses is characterized as a glass having the same constituent components, though delimited by the general requirement of maintaining an adequate crystallinity, clarity and elasticity for the glass. The rod and tube need not be formed from similar materials or even of materials from the same glass families. The materials are selected so as to have generally similar melting temperatures and CTE's such that during subsequent processing, the predominant mechanism for the mixing of materials between the rod and tube is interdiffusion rather than convection mixing. Generally, a melting temperature difference of within about 100 degrees Celsius and preferably within about 50 degrees Celsius is adequate. The coefficients of expansion between the rod and tube must be relatively closely matched to permit processing of the optical preform to temperatures at or above the softening points of the rod and tube glassy materials and with controlled slow cooling back to room temperature.

The separate choice of materials must also permit a rather finely polished surface to be created on the exterior cylindrical surface of the rod and interior bore surface of the tube. In order to create a uniform structure, voids of all kinds must be avoided at the interface between the rod and tube. In order to insert the rod into the bore of the tube, the radius of the rod must be slightly less than that of the bore.

Finally, the dimensions of the annular spacing between the rod and tube must be sized to account for any greater coefficient of thermal expansion by the rod relative to the tube, particularly where the melting point of the rod material is lower than that of the tube material. Ideally, the annular space is sized to permit the rod material to reach its melting point with a radial expansion sufficient to close the annular space between the rod and the tube without placing an expansive force on the interior surface of the tube in excess of the expansive strength of the tube. Once a glass rod has been mechanically inserted into the cylindrical bore of a tube, the composite structure is placed within a furnace that is substantially conventional in nature. The relevant modification to the furnace is the provision for ceramic rollers provided within the interior of the furnace to support and provide for the rotation of the composite preform. The composite structure is then heated to closely mate the bore surfaces and to eliminate any voids, annealed to relieve structural stresses and then cooled. The voids may also be eliminated by drawing the heated assemblage through a annular die This is a well known and practiced process. A glass carrier, constructed preferably of a high temperature resistant quartz glass tube, is used to encase the resulting composite preform. A non¬ stick agent such as finely powered boron nitride is used to prevent sticking of the glass to the quartz during subsequent thermal processing. The function of the glass carrier is to allow processing of the composite preform at temperatures above the softening temperature of both the rod and tube material. This high temperature processing is performed by placing the carrier, including the composite preform, into the furnace and onto the ceramic rollers. The temperature within the furnace is preferably raised at a conventionally determinable rate to a temperature in excess of the softening temperature of both the rod and tube. This higher temperature is selected to maximize the rate of interdiffusion of the rod and tube materials by interdiffusion, but without any substantial degree of convective mixing. In order to maximize this allowed temperature, minimize the likelihood of convective mixing due to uneven heating of the preform and to maintain the structural symmetry of the preform, the carrier is rotated beginning at a temperature significantly below the softening temperature of either the rod or tube material. The rate of rotation is preferably sufficient to allow the preform to be uniformly heated within the furnace. The rotation of the carrier is maintained for a period that will substantially define the thickness of the interface layer formed between the rod and tube. After formation of the interface layers, a controlled cooling of the carrier is performed down to an annealing temperature below the softening temperatures of both the rod and tube material as well as that of the deep bonded layer. This annealing temperature is, however, preferably selected to be sufficient to anneal the deeply placed interface layer. Rotation of the carrier by the ceramic rollers is preferably halted once this annealing temperature is reached. After a further annealing time, the carrier is cooled slowly to a final ambient temperature.

The first step in the process of the present invention is to choose a desired constituent density distribution appropriate for the required distribution of a physical property. For example, the optical designer may require a given spatial distribution of the index of refraction. Conventional design techniques can be used to determined the density distribution of a selected refractively active atomic species necessary to achieve the required spatial distribution of the physical property.

In accordance with the present invention, this prescribed atomic density distribution can be achieved through a process that considers a number of highly interdependently affected factors. These factors include the constituent materials of the core and tube materials and the concentrations of the active atomic species contained therein, the mobility of the active atomic species at the diffusion temperature, the viscosity of the glass materials at the diffusion temperature, the radial dimensions of the core and the number and radial dimension of each tube layer in the assemblage, the diffusion time and temperature, and the rotation rate of the assemblage about the central cylindrical axis of the core. The interdependency of these factors results in a highly non-linear system. Consequently, the system can be non-deterministic. However, the system can be made both deterministic and stable within a reasonable window of fabrication variation in the carrying out the process. In accordance with the present invention, the principle destablizing effects on the process have been determined to be the onset of hydrodynamic instabilities resulting in convection mixing of the constituent materials of the assemblage, the affect of the applied gravitational field vector on the assemblage in light of the differences in density of the constituent materials, and the affect of the gravitational field vector on the mobility of the active atomic species during diffusion. Other effects that must be accounted for include the non-infinite radial depth of the assemblage and the effect of rotation upon the concentration profile at the outer surface of the assemblage during diffusion. The problem of hydrodynamic instabilities can be avoided by reducing the factors that may give rise to turbulence within the assemblage. Decreasing variability in the co-axial rotation of the assemblage, decreasing the rotation rate applied during diffusion, and increasing the viscosity of the constituent materials through choice of composition or by reduced diffusion temperature will all serve to decrease hydrodynamic sensitivities within the system. The differential gravitational effect on the constituent materials of the assemblage due to differences in density can also be effectively nullified. If the difference in densities is not accounted for, the denser materials will tend to flow, initially creating a non-uniformity in the internal radial symmetry of the assemblage and ultimately a hydrodynamic instability that will result in a turbulent mixing of the constituent materials. The onset of a flow can be impeded by increasing the viscosity of the constituent materials at the diffusion temperature. Decreasing the difference in densities of the constituent materials, by choice of materials, will decrease the sensitivity of the system to the effects of gravity. Also, increasing the rotation rate of the assemblage during diffusion will tend to average the effects of the applied gravity vector. However, since the present invention fully contemplates the potential use of lower density constituent materials in outer tube layers of the assemblage, a too rapid rotation rate of the assemblage could result equally in centrifical force induced flows that similarly create non-uniformities and, ultimately, turbulence. Thus, the present invention provides for the controlled balance of factors to achieve an averaging of the applied gravitational force to zero.

The mobility of the active atomic species is dependent on both the atomic characteristics of the particular species and to the strength of the applied gravitational field. A mobility coefficient is defined as a measure of the drift velocity of a particular atomic species under the influence of a quantified external field, such as gravity. An exemplary mathematical description of atomic mobility, as well as atomic diffusion, is provided in "Diffusion in Solids" by P.G. Shewmon, McGraw-Hill 1963, page 203, which is expressly incorporated herein by reference. The mobility coefficient of the active atomic species used in the process of the present invention can be effectively controlled by choice of the particular atomic species utilized. However, as the choice of the active species is somewhat limited, the present invention provides for the averaging to zero of the drift velocity vector of the active atomic species through choice of the rotation rate of the assemblage during diffusion. Thus, in accordance with the present invention, an extension is made to the atomic diffusion equations in order to account for the effects of rotation and of gravity on the rotating assemblage.

Process for Identifying an Initial Assemblage and Parameters:

The method starts by selecting an initial rotation rate, temperature schedule, and an initial arrangement of an assemblage. This initial arrangement consists of an initial construct defined by the number of tubes used and the radii and materials of the tubes and core. This initial spatial distribution of the core and tubes establishes an initial distribution of the atomic species that is active in establishing the desired physical property. The method then determines, preferably through an iterative process, the evolution of the atomic species distribution through numerical calculations for a series of discrete time steps. After each time step, the calculated atomic distribution is compared with the desired constituent distribution using a quantitative measure of the difference, such as best fit or least mean squares. Our present teaching uses weighted least mean squares. The sequence of numerical calculations is repeated until a time step is reached where the difference measure is within a minimum tolerance range selected to define a stop limit for this series of time step calculations. The actual minimum difference measure, the temperature and rotation rate parameters of the process, the initial arrangement and materials of the assemblage, and the number of time steps required to realize the minimum, representing the elapsed diffusion time required, are recorded. The next step in the method provides for the choosing of a new initial arrangement of an assemblage or a change in a process parameter The numerical calculation sequence is then repeated and the results recorded. As sequences are completed the selection of different assemblages and changes to process parameters may be determined based on the results of the prior sequences. Thus, subsequent initial arrangements and process parameters are preferentially selected toward optimizing the next choice of initial arrangements and process parameters to yield the minimum possible difference measure. In addition to identifying the preferred initial assemblage, the method also determines the required diffusion time for the fabrication process The determined diffusion times can also be used as a parameter limit considered as part of the selection of process parameters and initial arrangements. Ultimately, an initial arrangement will be found that has a difference measure within the tolerances required by the optical design if the design is a feasible one. The initial arrangement and process parameters that produced the acceptable difference measurement can then be used to fabricate the radial lens blank.

The method of the present invention further includes the step of choosing the temperature-time schedule during the diffusion process to achieve desirable ends, such as minimizing the difference between the desired and achieved concentration distributions. The method further includes the step of choosing an appropriate rotation rate schedule for the process.

The method further includes the step of selecting an appropriate measure of the difference between the desired property distribution and that achieved by the diffusion from a given initial assemblage. This difference can include weights that take into account the tolerances of the optical design which may vary over the desired property distribution within the gradient element.

The physical coefficients and parameters of the constituent glasses that must be measured in order to utilize this method include the density of each relevant constituent, the coefficient of thermal expansion and the viscosity, the mobility coefficient, and the coefficient of diffusion as a function of the temperature.

An exemplary starting assemblage is illustrated in FIG.1 a. As shown, the concentric assemblage 10 consists of multiple cylinders or tubes with different but homogeneous compositions prepared to the point of encapsulated heating according to the method described by Ward-Pulsifer. The assemblage consists of a central solid rod 1 1 inserted into the hollow center of a tube 1 2, which in turn is inserted into a tube 13. The completed assemblage is then inserted into the high temperature quartz encasing tube.

In Fig. 1 b, the coordinate system used in the description of the present invention is shown relative to an end view of the cylindrical assemblage 10. The x and y Cartesian coordinates are shown as well as the radial coordinate r at an angle a measured from the x axis. The initial radial density distribution of the constituent of interest is illustrated in Fig. 2a. The assemblage 10 is then placed in the rotation mechanism and the rotation is started. The temperature is then raised and the assemblage is diffused. After a short diffusion time the sharp steps in the density distribution become rounded steps as illustrated in Fig. 2b. As the time increases, the distribution becomes smoother. Eventually the density distribution becomes very smooth as in Fig. 2c. Note that if the assemblage 10 were to be diffused for a very long time, the distribution would eventually become constant and the assemblage 10 would be homogeneous, assuming no evaporation from the surface. When the distribution achieves the desired profile, the temperature is lowered until the sample can no longer diffuse. Rotation is then stopped

In distinction from the prior art, the initial distribution of constituents need not be in a particular order. Ordinarily, denser materials must be placed on the bottom in a planar diffusion process so that the effect of gravity would not induce convection currents in the diffusing assemblage. However, the Ward-Pulsifer method allows the concentric tubes to be arranged in any order of relative density, since the effects of gravity are essentially averaged to zero. One possible initial radial density distribution in this situation is illustrated in Fig. 3a After a short period of interdiffusion, the sharp breaks in the density distribution become rounded as illustrated in Fig. 3b. As the diffusion time increases, the distribution becomes smoother. Eventually the density distribution becomes very smooth as in Fig. 3c. Note the near identity of the gradient profiles in Fig. 2c and Fig. 3c. This feature of the process, that the layers can be arranged in any order in density, gives the fabricator much more freedom in choosing the initial arrangement to minimize cost, achieve more general gradient profiles, or to simplify fabrication.

If the sample is not rotated, the heavier elements flow toward the bottom of the sample, destroying the radial symmetry of the element. If the rotation rate is appropriately chosen, the effects of gravity are averaged out and the radial symmetry can be preserved to high accuracy. However, if the rotation is too fast, convection currents and instabilities are excited and radial symmetry will be lost. The allowable rotation rates depend upon the viscosity throughout the assemblage at the temperature used for diffusion.

Extension to Several Diffusing Atomic Species: More than one diffusing atomic species may be advantageously used in certain applications and in order to achieve certain desirable characteristics. Using measurements of the densities of these species, the initial physical characteristics of interest can be computed and verified using known techniques. The diffusion of each atomic species can be computed as described above for each time step. The physical characteristic of interest, represented as a function of all of the density distributions, can then be calculated and compared with the desired value. The optimum set of initial parameters can then be selected from among the set of initial assemblages that were computed.

Theoretical Treatment:

The theory of the fabrication of radial gradients with prescribed profiles will now be described. The basic process utilizes the diffusion of atoms which have a strong interaction with light rays or electromagnetic waves into a solid material, the active atomic species. The general equation of stress and gravity-assisted diffusion is well established; see for example "Diffusion in Solids" by P.G. Shewmon, McGraw-Hill 1 963, Chapter 1 , page 25:

dC (r,a;t) = - V ^• J, dt where

J= - M(C,V (bVC(r,a;ϋ + C(r,a;tNV(r,a;t)l and the coordinates are defined and illustrated in Fig. 1 B. The initial condition is written as C(r,a;0) = C r), in which the radial symmetry of the initial assemblage is made explicit; there is no dependence on the angle a.

The constituent density C can be conveniently expressed in both Cartesian and cylindrical coordinate systems, C(x,y,z;t) = C(r,a,z;t). The cylindrical coordinates are ir,a,z) where the radial distance from the axis is r, the polar angle is a, and the distance along the axis is z. The geometry of the cylindrical assemblage insures that C(r,a;t) does not depend upon z. The parameter b is kT, where k is the Boltzmann constant and 7 is the temperature. The diffusion coefficient is given by D(C,T) = b M(C,T).

Finally, V(r,a;t) describes any potential field such as gravity that exerts a force on the diffusing atom or molecule.

For the rotation speeds of interest here, the equations can be written most easily in a coordinate frame that rotates with the assemblage. In this frame, the gravitational field can be written as

V(r,a;t) = mg [x cos(ft) + y sin(ft)],

where f is the rotation angular frequency, x = r cos(a), and y = r sin(a). The quantity m is the mass, or buoyancy, of the diffusing constituent relative to the inert constituents

In addition to the gravitational field, there is also a centripetal 'potential' that has the form

V r,a;t) = Δ ' mfr².

For the slow rotation rates utilized here, this term is small and will be neglected in the discussion here. It can be included in any exact numerical treatment. This equation can be treated by standard numerical techniques such as those described in "Numerical Recipes" by W.H. Press, et al., Cambridge University Press, 1986, Chap. 1 7, p. 635-640 and in "Computational Physics" by S.E. Koonin, Addison-Wesley, 1 986, pp. 1 66-1 71 . In order to illustrate the method and the preferred practice of this teaching, an exemplary case will be discussed.

If the mobility coefficient M is a function of temperature only, the equations can be simplified. The variable change

x ,, == x + g, (t), g, (0) = 0, Vt = Y + 9₂ fϋ. 9₂ ( ) = 0,

with the identification

g t)

= M (T) mg cos(ft) dt

dg₂ (t)

= M (T) mg sin(ft) dt

leads to the equation

where the initial condition is now written as C(x,,y,;0) = C₀(rj. It is convenient to solve this equation in cylindrical coordinates, in which

r = x + y v² _ ^_

¹ r_Λ r

and C{x y^t, - C(r; t).

This is a standard linear diffusion problem whose solution is well discussed in the mathematical and physics literature There are also many methods for solving this equation that are available in commercially available software packages.

In a more accurate treatment, the dependence of the mobility coefficient M(C, T) upon the concentration C(r,a;t) should be taken into account. The dependence of M(C, T) on C(r,a;t) can be measured by standard methods. Even though the detailed dependence varies among the different glass types, the general features of this teaching do not depend upon such details. However, the quantitative numerical results do depend upon this dependence. The differential equation becomes nonlinear but it remains stable and can be solved by standard methods. This problem can be used to motivate and to explain certain preferred embodiments of the method. It is preferred that the initial assemblage be set into rotation before the temperature is raised Thus, the functions g,(t) and g₂(t) are zero because MfT) is zero. The mobility coefficient depends strongly on the temperature. For the time period in which diffusion is taking place, the functions g,(t) and g₂(t) oscillate (equal positive and negative values). After the diffusion stage is completed, it is preferred that the temperature be lowered to stabilize the assemblage while the rotation is maintained. After this cooling period the functions g,(t) and g₂(t) are again constant. When the temperature is below the softening temperature, the rotation can be stopped. The time-temperature profile, which gives the time dependence of the mobility coefficient, determines the final values of g, and g₂. If the ramping up and ramping down of the temperature is sufficiently slow then the final values of g, (t) and g₂(t) are vanishingly small. The radial symmetry of the diffused assemblage is thereby preserved.

It is important to emphasize at this point that in the author's Patent No. 5,262,896 concerning the fabrication of axial gradient profiles, the ordering of the plates in the initial assemblage must be monotonic in density. In the present cylindrical case, since the effects of gravity are averaged to zero, the ordering of the densities of the concentric cylinders can be arbitrary.

Formulas that relate physical characteristics to the density of individual atomic species are common in the literature. For example, the use of the

Lorentz-Lorenz formula and the important formula given by M.L. Huggins is discussed in U.S. Pat. No. 5,262,896 by the author. All of these relations can be cast into the form

f(n(r,a;W - ffn,) - f(n_fJ = F fC(r,a;t) - C,],

where n, is an arbitrarily chosen index value, C, is its corresponding concentration value and F is a fitted constant. The time dependence of C is determined by the differential equation and its initial value/distribution is given by the above relation. Solving the above equation for n(r,a,t) will then yield its time evolution.

A numerical quantity is desired to reflect how close a particular sample is to the prescribed index profile. This quantity will be termed the measure of fit, or MOF. One possible way to quantify the measure of fit is to define the sum

MOF = ∑ iv. [n(r_ra_f-()- n (r_fa_f-t)f The sum over / is over a selected set of radius and angle values. The quantity w, is weight given by the optical designer and the quantity inside the square bracket is the difference between the sample index at n the point / and the prescribed index n_p at the point /. The smaller the numerical value of MOF the better the fit. The weights w, are chosen by the optical designer to emphasize those parts of profile which are important for the optical performance of the system. For example, most of the light energy enters a radial lens near its outer radius. Thus, the importance of the outer part of the index profile can be encoded by choosing larger weights for this region.

Computer Program Logic:

A computer program can be employed to select the appropriate parameters of the starting assemblage and the time-temperature script for the diffusion stage of the fabrication process. One such program is illustrated in Figure 4. The program is set up by a step 40, in which the desired index profile is input into the program along with the acceptable range of the MOF and its associated weights as prescribed by the optical designer. Since the index of refraction profile is to vary in the radial direction only, the input data can be input as a series of desired values of the index of refraction at specified values of the radius or as a mathematical formula with suitable parameters. The MOF can also be represented in either of these two different modes as the operator desires.

In a step 42, the different glass compositions available for the process and their physical characteristics are input into the computer. The physical characteristics that should be input include the index of refraction, the mobility and the diffusion coefficient at the operating temperature. Other physical characteristics needed are the coefficient of thermal expansion (or CTE) and the thermal conductivity. The specifics of a particular problem may require other charactenstics to be included, such as the chemical composition by weight or density fraction.

In a step 44, a starting construction for the initial assemblage made up of the available tubes is either input into the computer by an operator or the computer is allowed to generate an initial construction. In a step 46, the tube array is stored for further manipulation during the execution of the program. Next, the MOF is computed for this starting array and the starting concentration profile is also computed and stored in a step 48.

At this point the program logic enters into the loop that increases the time of diffusion, searching for the minimum MOF that is achievable from this particular starting assemblage and the time of diffusion that achieves this minimum. This loop is enclosed in a dashed rectangle in Figure 4. The first operation is a step 50, in which the time is set equal to zero. In a step 52 the time is incremented by a given amount and the differential equation is solved by one of the standard methods to yield the new concentration at the incremented time. From this new concentration, the associated index profile is evaluated and the new MOF is computed in a step 56.

In a next step 58, the new value of the MOF is compared to the previous value. If the MOF has improved, indicating an better fit to the desired profile, the program will store the MOF value and then repeat the loop, further increasing the time of diffusion. If the MOF has not improved, the program breaks out of the loop at step 58 and proceeds to a step 62, which checks whether or not the MOF is within the acceptable range. If it is acceptable, the initial array, final profile and diffusion time is stored in a step 64 and the computer passes to a step 66. If the MOF is not acceptable, the computer proceeds directly to step 66.

In step 66, the initial array is modified in a manner chosen by the operator and the computer reenters step 48 to search for other acceptable initial tube arrays. The type of modifications of the initial assemblage that can be utilized include modifying the thickness of the tubes, including tubes with intermediate index values, or eliminating tubes that are very thin since they will play a negligible role in determining the final index profile.

After a sufficient number of acceptable initial arrays have been generated by the computer, the operator may then proceed to choose one for actual fabrication. This choice may be based on, but not limited to, cost, accuracy, stability, optical, or chemical properties.

Many variations of the above teachings can be utilized in the application of the present invention without departing from this invention; the detailed descriptions are illustrations of the preferred embodiments and are not meant to be exhaustive.

Fabrication of Sample Radial Lenses:

Numerous samples of a radial lens element have been fabricated consistent with the present invention. These elements use an initial assemblage of a central core and one surrounding tube. The two glasses used are commercially available from Schott Glass Inc., of Duryea, PA. The cores were of type SF1 with an index of refraction of 1 .71 7 and diameters ranging from 2.6 to 4.0 millimeter. The tubes were of type F2 with a lower index of refraction of 1 .620 and external diameters ranging from 9.7 to 1 3.5 millimeter. All assemblages were heat drawn to the above dimensions to eliminate air gaps between the core and the tube. Following the Ward-Pulsifer process, the samples were encapsuled, rotated, and heated to temperatures ranging from 700 degrees C to 775 degrees C for periods of either 48 or 96 hours. The rate of rotation was either 29 or 52 rpm (revolutions per minute). All samples were found to have an imaging central region with diameters in the range 3-4 millimeter. The change of index across the imaging region varied, as expected, depending upon the processing time and temperature. The pitch of the radial lenses varied from 50 millimeter to 250 millimeter. When a light ray enters such a lens parallel to the optic axis, its path is sinusoidal, oscillating from one side of the centeriine to the other; pitch is defined as the travel distance required for the light ray to return to its original distance from the axis. For example, one lens was fabricated from a core with a diameter of 4.0 millimeter. The tube had an outside diameter of 1 3.5 millimeter. It was processed at a temperature of 725 degrees C for 48 hours at 52 rpm. Its length was 26.2 millimeter which yielded a one-quarter pitch lens. That is, all light rays entering one end parallel to the optical axis exited the other end at the center of the lens. The focusing region of the lens had a diameter of 3.4 millimeter and its index gradient constant, -JA, was measured to be 0.060 per millimeter. All measurements on the sample lenses were consistent with the solution to the differential equation given in the theory section for a selected value of the mobility constant M(T). The temperature dependence of the mobility constant was also evident in the data from the various samples.

A summary of this and six similarly processed samples is provided in the table below:

Sample Temperature Rotation Hours Diameter of VA Number Optical Region Measured

1 700 48 3 2mm 0 055

2 700 48 3 Omm 0 075

3 725 48 3 4mm 0 063

4 750 48 3 3mm 0 048

5 760 48 3 1 mm 0 038 β 760 96 3 6mm 0 025

7 775 48 2 7mm 0 034

An eighth sample was fabricated to explore the effects of rapid rotation at a high temperature with the resultant low viscosity of the glass in the assemblage. The core diameter of the sample was measured at 3.4 millimeters with an outside diameter measured at 10.5 millimeters. This sample was processed at a temperature of 800 degrees Centigrade for 32 hours at a rotation rate of 201 revolutions per minute. The resultant sample demonstrated the calculated amount of diffusion, but the central imaging region had optically deformed in a corkscrew shape with an offset averaging approximately 2.7 millimeters from the central axis of the sample. The wavelength of the central imaging region was measured to have a length of approximately 10 to 1 1 millimeters. The high rotation rate is believed to have induced the onset of a hydrodynamic instability within the sample resulting in a physical flow of the central imaging region material.

In view of the viscosity of the materials at the temperatures applied to the sample during rotation and the duration of the diffusion period, a reasonable upper limit on the rotation rate for this sample without incurring a hydrodynamic instability is about 1 75 revolutions per minute. Thus, a method of predictably fabricating radial gradient preforms has been described. The method is particularly applicable to preforms formed through the use of the Ward-Pulsifer method. The method of the present invention, however, is also applicable to the formation of radial gradients in other materials and to use in other systems presenting different and additional processing steps to obtain a radial gradient of a physical property through an interdiffusion of materials.

Accordingly, various modifications and variations of the present invention may be made by those skilled in the art within the scope of and without departing from the teachings of the present invention as set forth in the appended claims.

Claims

1 . A process of fabricating a preform having a predetermined radialprofile defining a spatial distribution of an active atomic species, said process comprising the steps of: a) choosing a preform blank including an assemblage of materials including respective concentrations of an active atomic species; b) determining a set of process parameters including time, temperature and rotation rate for application to said assemblage that will yield a minimum deviation from a predetermined final radially symmetric spatial distribution of said active atomic species within said assemblage, said process parameters being determined to provide for controlled interdiffusion of the materials within said assemblage while effectively maintaining a zero average applied gravitational field and zero average atomic mobility of the active atomic species relative to said active atomic species within said assemblage; and c) using said chosen preform blank and said set of process parameters to form a final preform structure substantially having said predetermined final radially symmetric spatial distribution of said active atomic species within said assemblage.

2. The process of Claim 1 wherein said step of determining further provides for said process parameters being determined to provide for controlled interdiffusion subject to the existence of a diffusion barrier at the outermost surface of said preform blank.

3. The process of Claim 1 or 2 wherein said step of determining further provides for said process parameters being determined to provide for hydrodynamic stability of said materials within said assemblage by interdependent selection of the viscosity of said materials within said assemblage, the temperature of said assemblage, and the rotation rate applied to said assemblage during diffusion

4. A method of making a refractive element having a radially symmetric spatial distribution of a physical property comprising the steps of:

(a) determining a final spatial distribution of an active atomic species within an inert optical matrix that corresponds to a predetermined spatial distribution of a predetermined physical property;

(b) selecting one of a plurality of initial radially symmetric spatial distributions corresponding to an arrangement of physical materials formed from said inert optical matrix and including said active atomic species;

(c) selecting a profile of time, temperature, and rotation rate for application to said arrangement;

(d) compute an intermediate spacial distribution of said active atomic species in said arrangement for said profile;

(e) compute a difference measure between said intermediate and final spacial distributions;

(f) selecting a particular one of said plurality of initial radially symmetric spatial distributions and a corresponding profile having a minimum difference measure; and

(g) rotating and diffusing said arrangement of physical materials having said initial radially symmetric spatial distribution subject to said corresponding profile.

5. The method of Claim 4 further comprising the steps of

(g) selecting from the set in step (c) one theoretical arrangement that best satisfies a suitably chosen criteria such as minimum cost or maximum optical performance; (e) fabricating glass cylinders with parameters as given by the theoretical arrangement chosen in step (d), thereby forming the starting assemblage with radial symmetry;

6. A method as claimed in claim 4 wherein the given physical property is index of refraction.

7. A method as claimed in claim 4 wherein the given physical property is the coefficient of thermal expansion.

8. A method as claimed in claim 4 wherein said time-temperature profiles consist of a time interval during which the temperature increases, an interval of preselected constant temperature and a final interval during which the temperature decreases and the three time intervals vary from instance to instance.

9. A method as claimed in claim 8 wherein said time- temperature-rotation profiles consist additionally of a preselected rotation rate.

10. A method as claimed in claim 4 wherein said time-rotation profile consists of an increasing rotation rate during the interval of increasing temperature, a preselected constant rate during the interval of constant temperature, and a decreasing rotation rate during the interval of decreasing temperature.

1 1 . A method as claimed in claim 4 wherein said starting spatial distribution consists of a series of closely fitting concentric cylinders with each cylinder having homogeneous physical properties. The central cylinder consists of a solid rod.

1 2. A method as claimed in claim 1 1 wherein each of the said starting concentric cylinders has a value of the physical property which is different from that of its immediate neighboring cylinders.

1 3. A method as claimed in claim 1 1 wherein all of the said starting concentπc cylinders are chosen from samples that have only two different values of the physical property.

14. A method as claimed in claim 1 1 wherein the said starting concentric cylinders are not necessarily ordered monotonically in their density.