Segmented axial gradinet array lens

A segmented axial gradient array lens is provided. In one embodiment, the segmented array lens includes two sheets of optical material which mate at opposing faces which have a series of corresponding parallel grooves and ridges. At least one of the sheets has an axial gradient index of refraction profile. This provides the functionality of a series of parallel cylindrical lenses. The array lenses may include one or more additional intermediate sheets which may have a homogeneous or axial gradient index. With the first and last sheets having an axial gradient index of refraction, and the grooves forming the first and last interfaces between the sheets being rotated 90 degrees relative to each other, a two dimensional array of point foci can be provided. An array of cones and conical indentations can be used at the interfaces to provide the functionality of an array of spherical lenses. An array lens of the invention can be used to provide an optical multiplexer.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention relates generally to lenses, and more particularly to 
array lenses formed from a plurality of sheets of optical material, some 
of which have axial refractive index gradients. 
2. Prior Art 
A conventional lens with spherical surfaces and with a homogeneous index of 
refraction will not focus light perfectly; there will be spherical and 
chromatic aberrations. The latter aberrations may be canceled, for 
example, by using a lens doublet in which each lens has a unique chemical 
composition and therefore an index of refraction with its own dependence 
on the wavelength of light. The chromatic aberrations can be reduced by 
cancellation between the two indices. The spherical aberrations can be 
eliminated by the expensive and difficult process of grinding a 
predetermined aspherical surface on the lens. It is also well known in the 
art that these aberrations can be eliminated by employing axial gradient 
lens blanks. An axial gradient lens is a lens which has an index of 
refraction profile which varies in one direction only, usually chosen to 
be the optical axis. These aberration-free lenses can be used 
advantageously in a variety of optical systems, such as slide projectors, 
cameras, binoculars, and many other imaging devices; the number of lens 
elements required for a given task can be reduced as well as the weight 
and complexity of the system. 
The blanks for the fabrication of such gradient lenses can be made by a 
variety of processes such as SOL-GEL, infusion, and diffusion and may be 
glass, plastic or other suitable optical material. In particular, there is 
the controlled diffusion process that can produce macro lenses with a 
prescribable index of refraction axial profile. The fabrication of such 
axial gradient lenses by the controlled diffusion process is disclosed in 
U.S. Pat. No. 5,262,896, "Refractive Elements With Graded Properties and 
Methods for Making Same", to R. Blankenbecler, which patent is 
incorporated herein by reference. These lenses are available from 
LightPath Technologies of Tucson, Ariz. 
The above discussion applies to both radial and cylindrical lenses; however 
the grinding and polishing of cylindrical lenses to the needed precision 
is especially difficult. Cylindrical lenses condense or expand a beam of 
light in one transverse dimension only; they can focus light into a thin 
line and are used in laser scanners, fax machines, laser printers, and in 
the Cinemascope process, for example. Cylindrical lenses also can be used 
to produce a symmetrical output beam from an unsymmetrical source such as 
a laser diode. 
A lens design for coupling a laser diode to a multimedia fiber using 
anamorphic radial gradient-index components is described by J. M. Stagaman 
and D. T. Moore, "Laser diode to fiber coupling using anamorphic 
gradient-index lenses", Applied Optics, vol. 23, no. 11, pp. 1730-1734 
(1984). These authors discuss the disadvantages and difficulties in the 
conventional approach of utilizing prisms and/or cylindrical lenses. Their 
optimum design for a lens system to be used with a laser diode source with 
astigmatism uses a gradient index lens with an elliptical profile. 
However, there is no known method to fabricate a general anamorphic lens 
in which each transverse dimension has its own independent index profile. 
A monolithic anamorphic lens having at least one curved surface and an 
axial gradient index parallel to an optical axis is disclosed in U.S. Pat. 
No. 4,900,138 to Atkinson, III, et al., issued Feb. 13, 1990. This patent 
also describes other gradient index profiles and is incorporated herein by 
reference. 
A cemented lens design in which two (or more) different types of 
homogeneous glass are ground into the proper shape then polished and 
cemented together, is well known in the art. In Applied Optics, by Leo 
Levi, Vol. 1, John Wiley & Sons, New York, (1968), it is stated that with 
a proper choice of glasses and shapes, it is possible to reduce both 
chromatic and spherical aberrations in a cemented lens despite the severe 
restrictions on available choices. A variation of this type of lens has 
also been described by A. C. S. van Heel in "One-Radius Doublets", Optica 
Acta, Vol. 2, pp. 29-35 (1955). 
Also known to the art is a segmented lens design in which two different 
types of homogeneous glass with different values of the index of 
refraction are bonded together by heat and the interface molded (slumped) 
into a prescribed shape. The external faces are then ground flat. This is 
a monolithic lens with a discontinuous index of refraction across a smooth 
aspherical interface. This lens has low optical power and has been 
proposed as a corrector plate. Such a lens is disclosed in U.S. Pat. No. 
2,596,799, "Aberration Corrector Member for Image Forming Optical 
Objectives", to Tillyer, et. al., issued May 13, 1952. 
Another type of segmented lens known to the art is built up of constituents 
of different indices of refraction as described by W. Ewald, in "Lens for 
Optical Purposes" U.S. Pat. No. 1,943,521, issued Jan. 16, 1934. The 
separate parts of the lenses, each of which is homogeneous, are cemented 
together in such a manner that the boundary surfaces or interfaces are 
substantially located in the direction of the path of light rays. That is, 
the interfaces are parallel to the optical axis. The indices of refraction 
are chosen so as to reduce the spherical aberration of the lenses and 
produce clearly defined images on a screen. 
A patent teaching the fabrication and design of a double axial gradient 
lens blank has been granted to the present inventor, R. Blankenbecler, 
"Double Axial Gradient Lens and Process for Fabrication Thereof", U.S. 
Pat. No. 5,044,737, issued Sep. 3, 1991, which patent is incorporated 
herein by reference. A diffusion process produces a monolithic lens with a 
continuous index of refraction profile; the lens is composed of three 
regions, front, center, and rear, each of which can have its own graded 
index of refraction profile and chemical composition. 
A patent teaching the forming of a cylindrical or spherical gradient lens 
blank from an axial gradient lens blank by heat molding (slumping) has 
been granted to R. Blankenbecler and M. Wickson, "Shaped Gradient 
Fabrication In Lenses By Molding From Axial Gradient", U.S. Pat. No. 
5,236,486, issued Aug. 17, 1993, which patent is incorporated herein by 
reference. This process produces a monolithic lens with a continuous index 
of refraction profile. 
A design for a cemented gradient index lens system for laser beam reshaping 
is disclosed by C. Wang and D. L. Shealy, "Design of gradient-index lens 
systems for laser beam reshaping", Applied Optics, vol. 32, pp. 4763-4769 
(1993). A system using two axial gradient lenses and a homogeneous central 
transfer lens is disclosed. The front and rear faces are flat planes. The 
interfaces between the front gradient lens and the central transfer lens 
and the central transfer lens and the rear gradient lens are spherical 
surfaces that must be ground and polished to fit into each other. In 
addition, the gradient index profiles are different and must be chosen 
properly to function as a beam reshaper. 
As mentioned above, spherical and chromatic aberrations will be present in 
lenses with spherical or cylindrical external surfaces. A suitable index 
of refraction profile (essentially linear) in the spherical or cylindrical 
lens cap can be used to cancel the spherical aberration and form a precise 
image. However this normally requires a large change in index across the 
profile. Furthermore, the surface of the cap must be ground and polished 
in a region of varying index, varying hardness, and varying coefficient of 
thermal expansion which is a difficult process to carry out with accuracy. 
An aspherical shaped surface on a homogeneous lens will also reduce this 
particular aberration, but the fabrication of such a lens surface with the 
required accuracy is a very difficult process. Even the grinding and 
polishing of a simple cylindrical surface on a lens blank is difficult and 
expensive in comparison to a spherical surface. 
There are many applications in optics for a lens array that breaks up an 
incident beam into many smaller beams and focuses each of these onto a 
separate focal point. One example of such a system would involve a 
detector composed of many separate pixel elements at the foci. The 
fabrication of a lens array composed of many small lenses that must be 
accurately assembled and aligned is both difficult and expensive. It would 
be desirable to provide an optical array that is simple, rugged, and easy 
to assemble which would focus a light beam onto an array of focal points. 
A multiplexer is a device which takes a single input signal and splits it 
into a plurality of identical output signals. Multiplexers are common 
components in electronic and microwave devices but acceptable multiplexers 
are not available for optical systems. It would be desirable to have a 
simple, rugged, and easy to assemble multiplexer for use with optical 
devices and processing systems. 
It is therefore an object of the invention to provide an improved type of 
lens having the function of an array of cylindrical or spherical lenses 
but which does not require individual lens elements or cylindrical or 
spherical surfaces anywhere in the lens. 
It is another object of the invention to provide a lens for use as an 
optical multiplexer. 
It is still another object of the invention to provide an array lens system 
which provides independent manipulation of a beam of light in two 
transverse orthogonal directions to provide a two dimensional output array 
of foci. 
SUMMARY OF THE INVENTION 
The segmented axial gradient array lens of the invention provides a 
flexible framework for the lens designer to achieve many desirable optical 
functions. In particular, an array lens having the functionality of an 
arrangement of difficult to fabricate cylindrical or spherical lenses is 
provided. 
In a preferred embodiment of the invention, a segmented array lens having a 
front surface, a rear surface and a plurality of parallel optical axes is 
provided. The lens includes first and second planar sheets of optical 
material such as glass or plastic. The first planar sheet includes first 
and second sides and has an axial gradient index of refraction profile. 
The second side of said first sheet has a series of parallel grooves which 
are each perpendicular to one of the optical axes, each of the grooves 
including a pair of planar walls with adjacent ones of the grooves 
intersecting at one or more ridges parallel to the grooves. The second 
planar sheet has first and second sides with the first side of the second 
sheet including a series of corresponding parallel grooves with each of 
the grooves including a pair of corresponding planar walls. Adjacent ones 
of the grooves intersect at one or more ridges parallel to the grooves. 
The ridges of the second side of the first sheet mate with the grooves of 
the first side of the second sheet and the ridges of the first side of the 
second sheet mate with the grooves of the second side of the first sheet 
to form a continuous interface with no air gaps from the first sheet to 
the second sheet. At least one wall of the grooves forms a finite, 
non-normal angle with an optical axis. If one of the sheets has a suitably 
chosen axial index of refraction profile, this configuration provides an 
array lens which takes an input of a light beam and produces an output of 
a series of line foci. This has the functionality of an array of parallel 
cylindrical lenses. 
In an alternative equally preferred embodiment of the invention, a 
segmented array lens including a plurality of planar sheets is provided. 
The lens has a front surface, a rear surface and a plurality of parallel 
optical axes and each sheet has a first and a second side. The first side 
of a first sheet forms the front surface of the lens and the second side 
of the first sheet has a series of parallel grooves, each of the grooves 
including a pair of planar walls, adjacent ones of the grooves 
intersecting at one or more ridges parallel to the grooves. The second 
side of a last sheet forms the rear surface and the first side of the last 
sheet includes a series of parallel grooves, each of the grooves including 
a pair of planar walls, adjacent ones of the grooves intersecting at one 
or more ridges parallel to the grooves. The lens further includes one or 
more intermediate sheets positioned between the first and second sheets. 
The first and second sides of each of the intermediate sheets each has a 
series of parallel grooves. The ridges and grooves of adjacent ones of the 
sheets mate to form continuous interfaces between adjacent sheets. At 
least one wall of the grooves forms a finite, non-normal angle with an 
optical axis. In one example of this embodiment there are three sheets 
with the first and last sheets having an axial gradient index of 
refraction and the single intermediate sheet having a homogeneous index of 
refraction. If the grooves between the first and intermediate sheets are 
rotated 90 degrees about a central axis relative to the grooves forming 
the interface between the intermediate and last sheets, the output of the 
array lens is a two dimensional array of point loci at the image plane of 
the lens. If these grooves are instead parallel, the output is a series of 
parallel beam strips. Additional intermediate sheets can be utilized to 
provide desired magnification and focal length properties. 
In another alternative embodiment of the invention, rather than grooves at 
the interfaces between sheets, arrays of cones and conical indentations 
are utilized. Each cone functions as a spherical lens and the output of 
the array lens is a two dimensional array of point foci. 
The array lenses of the invention may be used to provide optical 
multiplexers. This will take a relatively narrow input beam, such as would 
be provided by an optic fiber and splits it into two or more parallel 
output beams. This is preferably provided using two sheets of optical 
material with the first sheet having an axial gradient index of refraction 
and the second sheet having a homogeneous index of refraction. The number 
of output beams is governed by the number of ridges at the interface 
between the two sheets. To achieve this functionality, the input beam must 
diverge within the first sheet by either using the natural divergence of 
the beam or by providing a diverging stage at the input of the first 
sheet. Such an optical multiplexer will find use in optical processing and 
in connections in optical fiber networks. 
In each of the embodiments, the sheets are clamped, cemented, heat bonded, 
or otherwise joined using conventional technology to form a monolithic 
lens assembly. 
The segmented axial gradient array lens offers increased flexibility to the 
optical designer. Since the optical power of the lens is provided by the 
difference in index between the adjoining segments, the lens can exhibit 
smaller chromatic aberrations than a conventional lens. The chemical 
composition of each sheet can be chosen to ameliorate the chromatic 
aberrations.

DETAILED DESCRIPTION OF THE INVENTION 
The basic principles of segmented axial gradient lenses will first be 
discussed with reference to FIGS. 1-9 and then the segmented axial 
gradient array lenses of the invention will be described with reference to 
FIGS. 10-18. As used herein, a segmented lens is a lens composed of two or 
more close-fitting segments or sections. An array lens expands this 
structure using sheets of optical material and providing a two dimensional 
array output. The contact surface between two segments or sheets is an 
interface. In addition, there is a front external surface and a rear 
external surface of the complete lens assembly. At least one of the 
segments or sheets contains an axial gradient in its index of refraction 
with the remaining segment or sheet having a homogeneous index of 
refraction or an axial gradient index. The chemical composition of each 
segment or sheet may be chosen independently subject only to being 
consistent with the desired optical performance. The final lens assembly 
has the segments or sheets clamped, cemented, heat bonded or otherwise 
joined to form a monolithic unit. One novel feature of the invention is 
that the interfaces between the sheets comprise either grooves or cones. 
At least one wall of the grooves is set at a finite, non-normal angle to 
the optical axis. For purposes of this application, this is intended to 
mean that a general ray drawn parallel to the optical axes will intersect 
the interface (excepting the ridges) at a non-normal angle. In addition, 
that wall is non-normal to the optical axis. These simple surfaces are 
easily fabricated using standard techniques well known in the art. No 
spherical or aspherical interface surfaces are required. 
Another novel feature of the invention is that the optical power and the 
optical function of the lens is provided by prescribing the index of 
refraction profile in the various segments. The difference in index 
between the adjoining sheets produces a desired bend in an incident light 
ray path if it strikes the interface at a non-normal angle. Thus the 
interfaces are set at a finite angle to the optical axis. The resultant 
path of the light ray can be calculated from known laws of optics. 
A standard homogeneous cylindrical lens 10 of plano-convex type with an 
optical axis 12 is depicted in FIG. 1 as a single representative of the 
prior art. A segment 14 has a constant index of refraction throughout. A 
rear surface 16 is cylindrical and a front face 18 is a plane with the 
rear surface 16 functioning to focus light in a single transverse 
direction. 
In FIGS. 2A and 2B, the simplest embodiment of a segmented axial gradient 
lens is illustrated. A two segment cylindrical lens 20 is shown with an 
optical axis 12 in an exploded view in FIG. 2A and an assembled view in 
FIG. 2B. This design has a plane interface 22, a plane front surface 24 
and a plane rear surface 26. A first segment 28 has an appropriately 
chosen axial variation in its index of refraction while a second segment 
29 is homogeneous. In all the figures, the lines on the sides of the 
segments mark the planes of constant index of refraction which are normal 
to the optical axis 12. For a fixed total variation in the index of 
refraction along the interface, i.e., as measured down the slope of the 
interface 22, the focal length increases as the interface plane moves 
toward perpendicular to the optical axis 12. The image which exits the 
lens 20 can be moved perpendicular to the optical axis 12 (vertically in 
FIG. 2) by changing the value of the index of refraction of the 
homogeneous region. 
The use of a segmented axial gradient lens 30 with the cylindrical external 
surface 16 is illustrated in FIG. 2C. When the index of refraction profile 
of segment 28 is chosen appropriately, the lens functions as a compound 
lens and, in addition, the aberrations arising from the constant curvature 
of the surface 16 of the homogeneous segment 34 can be canceled. The focal 
length given by the curved surface 16 can be preserved and the aberrations 
still canceled by choosing an axial index profile in the front segment 28 
which varies slowly, essentially as the cube of the distance from the 
point where the optical axis 12 crosses the interface 22. Thus the curved 
surface 16 is fabricated in the homogeneous segment 34, a well known art, 
while the axial gradient segment 28 only has one flat front face 24 and 
one flat interface 22. 
Homogeneous segment 29 in FIGS. 2A and 2B can be replaced by a segment 36 
with an axial gradient region as depicted in FIG. 2D to provide a lens 38. 
If the index profile in segment 36 varies opposite to the index profile of 
segment 28, the lens 38 having a cylindrical functionality will have a 
shorter focal length (for the same total change in index of refraction in 
each segment) for appropriately chosen profiles. It is the difference in 
the value of the index of refraction in the two profiles at the point of 
intersection of a light ray with the interface 22 that determines the ray 
paths. Thus there is more freedom in choosing the individual profiles in 
the embodiment of FIG. 2D in that a designer has a larger set of 
parameters to vary in order to optimize selected optical and physical 
properties of the lens 38. 
The plane interface 22 of FIGS. 2A-2D can be replaced by a wedge or cuneate 
interface as shown in FIGS. 3A-3D. An exploded view of a lens 40 is shown 
in FIG. 3A, and an assembled view is shown in FIG. 3B. A first wedge 
segment 42 with a linear apex or ridge 44 has an appropriate variation in 
its index of refraction while a second segment 46 is homogeneous. The 
index of refraction of homogeneous segment 46 is preferably, but not 
necessarily, chosen to be equal to the value of the index of refraction at 
ridge 44 of segment 42. It is obvious that the lens 40 will also function 
if segment 42 is homogeneous and segment 46 has a suitably chosen axial 
gradient. The use of an axial gradient wedge segment 42 and a homogeneous 
segment 48 with a cylindrical external surface 16 is illustrated with a 
lens 50 in FIG. 3C. If the index of refraction profile of segment 42 is 
chosen appropriately, the lens 52 will function as a compound lens and the 
aberrations arising from the surface curvature of segment 50 can be 
canceled. The focal length given by the curved surface can be preserved 
and the aberrations canceled by choosing an axial index profile in the 
wedge segment 42 which varies slowly, essentially as the cube of the 
distance from the point where the optical axis 12 crosses the wedge 
interface 44. In this embodiment, the curved surface 16 fabricated in the 
homogeneous segment, a well known art, while the axial gradient segment 42 
only has a flat front surface and flat interfaces. As in the case of the 
plane interface, homogeneous segment 46 can be replaced by a segment 54 
with a suitably chosen index gradient to yield a more flexible cylindrical 
lens functionality as is shown in FIG. 3D. 
The linear wedge in the above embodiments can be replaced by a cone 
geometry as shown in FIGS. 4A-4D. An exploded view of lens 60 is shown in 
FIG. 4A and an assembled cross-sectional view is shown in FIG. 4B. A cone 
segment 62 with an axial gradient profile and a pointed tip 64 is inserted 
into an appropriately shaped homogeneous segment 66. The resultant lens 60 
has a radial symmetry. The index of refraction of homogeneous segment 66 
is preferably chosen to be equal to the value of the index at tip 64 of 
segment 62. When the index of refraction profile of segment 62 is chosen 
appropriately, the lens will produce a point focus instead of the line 
focus characteristic of a cylindrical lens. The use of an axial gradient 
cone segment 62 and a homogeneous segment 70 with a spherical external 
surface is illustrated with a lens 74 in FIG. 4C. If the index of 
refraction profile of segment 62 is chosen appropriately, the lens 74 will 
function as a compound lens and the aberrations arising from the surface 
curvature of segment 70 can be canceled. The focal length given by the 
curved surface can be preserved and the aberrations canceled by choosing 
an axial index profile in the wedge segment 62 which varies slowly, 
essentially as the cube of the distance from the point where the optical 
axis 12 crosses the cone interface point 64. In this embodiment, the 
curved surface is fabricated in the homogeneous segment, a well known art, 
while the axial gradient segment 62 only has a flat front surface and cone 
shaped interface. As illustrated in FIG. 4D, the homogeneous segment can 
be replaced by a segment 76 with a gradient index profile that varies 
opposite to that of segment 62 to yield a more flexible design for a lens 
78. 
Other desirable optical functions can be achieved by a three segment lens. 
A lens having three segments, each with its own independent index of 
refraction profile, will give the lens designer the maximum flexibility. 
However, for simplicity, several examples in which one of the segments, 
the central or middle one, is homogeneous in the index of refraction will 
be described. 
FIGS. 5A-5C show several three segment lens designs with two internal 
interfaces. In FIG. 5A, a lens 80 is depicted which is formed from an 
axial gradient segment 28 followed by a homogeneous segment 82 which in 
turn is followed by a second gradient segment 84 to form lens 80. This 
design has a front plane interface 86 and a rear plane interface 88. These 
segments have only plane optical surfaces and are cut to fit into each 
other. In FIG. 5B, a lens 90 with two optical wedge interfaces is 
depicted. An axial gradient wedge segment 62 is followed by a homogeneous 
segment 92 which in turn is followed by a second gradient wedge segment 
94. These segments have only plane wedge interfaces and are cut to fit 
into each other. Homogeneous segment 92 can be replaced by a tapered 
segment 96 as shown in FIG. 5C. Both of these designs have a wedge shaped 
interface 98 in the rear and a wedge shaped interface 99 in the front as 
depicted in FIG. 5B. 
If the wedges are aligned parallel to each other as in FIG. 5B and FIG. 5C, 
the index of refraction profiles in the first and third segments can be 
chosen so that the assembly operates as a cylindrical convex-convex, 
convex-concave, concave-convex or concave-concave lens. By appropriate 
choice of the index profiles, the resultant lens can be fabricated as 
either a positive (convergent) or a negative (divergent) cylindrical lens. 
A cylindrical beam expander/contractor can be achieved by using the 
three-segment design types shown in FIGS. 5A-5C. The index of refraction 
profiles in segment 28 and segment 84 in FIG. 5A, and the index profiles 
in segment 62 and segment 94 in FIGS. 5B and 5C, can be chosen to scale 
the beam in one transverse direction while keeping the incident and exit 
rays parallel. The scaling of the beam can be either a contraction or an 
expansion in one dimension. 
A cylindrical beam reshaper can also be achieved by the same design layout 
as illustrated in FIGS. 5A-5C. The index profiles in segment 28 and 
segment 84 in FIG. 5A, and the index profiles in segment 62 and segment 94 
in FIGS. 5B and 5C, are chosen both to scale the beam and to redistribute 
the beam intensity as the designer requires. For example, in certain 
applications it is desirable to reshape the beam from a laser from an 
(essentially) gaussian transverse intensity distribution into a 
substantially flat distribution. This can be accomplished by choice of the 
index of refraction profiles of the front and rear axial gradient 
segments. 
Alternatively, the wedges can be aligned perpendicular to each other, as 
shown with a lens 100 in FIG. 6. An axial gradient wedge segment 62 is 
followed by a homogeneous segment 102 which is followed by a second 
gradient wedge segment 104 which is rotated 90 degrees from the 
orientation of wedge segment 62. The index of refraction profile in 
segment 62 and in segment 104 can be chosen so that the assembly 100 
operates functionally as two cylindrical lenses at right angles to each 
other. By appropriate choice of these two index profiles, the resultant 
lens can function as a general as anamorphic lens in which the optical 
parameters of the two transverse directions can be chosen independently. 
By placing two of the cylindrical function three-segment axial gradient 
lenses as described above, one behind the other but rotated 90 degrees 
about the optical axis, a beam can be independently manipulated in the two 
orthogonal transverse directions. Three lens systems utilizing this 
configuration are shown in FIG. 7. In FIG. 7A, two cylindrical lenses of 
the type depicted in FIG. 2B are arranged around optical axis 12. A 
segmented axial gradient lens 110 which will bend light in the vertical 
direction only is followed by a similar lens 112 which is rotated 90 
degrees about optical axis 12 so that it bends light only in the 
horizontal direction. In FIG. 7B, two cylindrical lenses of the type 
depicted in FIG. 3B are arranged around optical axis 12. A segmented axial 
gradient wedge lens 120 which will bend light in the vertical direction 
only is followed by a similar lens 122 which is rotated 90 degrees so that 
it bends light only in the horizontal direction. In FIG. 7C, two 
three-segment lenses of the type depicted in FIG. 5B are arranged around 
optical axis 12. A three-segment axial gradient lens 130 which will affect 
the light rays only in the vertical direction is followed by a similar 
lens 132 which is rotated 90 degrees so that it affects the light rays 
only in the horizontal direction. 
The optical power of the segmented axial gradient lens is due to the 
difference in index of refraction between the adjoining segments. The 
chemical composition of each segment can be chosen independently such as 
to ameliorate the chromatic aberrations in analogy to a standard chromatic 
doublet. The segmented axial gradient lens offers increased flexibility to 
the optical designer in its geometric parameters and the chemical 
composition of each segment. It also offers simple surfaces to the lens 
fabricator and simple index of refraction profiles to the maker of the 
lens blanks. 
Theoretical Treatment 
The following theoretical treatment for segmented axial gradient lenses 
provides the basics for selecting various lens parameters of a segmented 
axial gradient array lens of the invention. The basic theoretical formulae 
given below are exact, but their solution will be given only in the small 
angle approximation. These are meant to demonstrate the general overall 
parameters and index of refraction profile required for the segmented 
axial gradient lens. Exact calculations of the properties needed to 
achieve a required performance of such a lens can be performed by several 
commercially available optical design software packages. One such package 
is "Code V" available from Optical Research Associates of Pasadena, Calif. 
Other packages are "Synopsis" from BRO, Inc. of Tucson, Ariz. and "ZEMAX" 
from Focus Software, Inc. of Tucson, Ariz. The coordinates used in the 
discussion are defined and illustrated in FIGS. 8A and 8B for the 
segmented axial gradient lens embodiments illustrated in FIGS. 3B and 5B 
respectively. 
For the two segment wedge lens illustrated in FIG. 8A, the equations that 
determine the optical ray path for a horizontal ray displaced a distance 
of y above the central optical axis 12 are 
EQU n1(z) sin (A1)=n1(0) sin (A1+B) (1) 
EQU n1(0) sin (B)=na*sin (C) (2) 
EQU y=(F-L) tan (C)+(L+z) tan (B) (3) 
EQU z=y*tan (A1)=y*z1/y1 (4) 
where: 
n1(z) is the index of refraction of segment 42 as a function of the 
distance z from the apex 44 of the wedge segment 42 back along the optical 
axis 12 toward the front face of segment 42; 
A1 is the base angle for the wedge; 
n1(0) is the index of refraction of homogeneous segment 48 (which is 
defined here to be equal to the index at the apex of the wedge); 
B is the angle a light ray is bent at the interface; na is the index of 
refraction of the medium within which the lens is positioned; 
C is the angle at which a ray exiting the rear face of the lens is bent; 
F is the focal length of the lens from the apex of the wedge; 
L is the distance from the apex of the wedge to the rear surface; 
z1 is the distance from the front face of the lens to the apex of the 
wedge; and 
y1 is one half the width of the base of the wedge. 
A similar and symmetric treatment holds for rays incident below the central 
optical axis. Equations (1) and (2) are Snell's law of refraction applied 
at the wedge interface and at the rear surface. Equations (3) and (4) are 
geometric. For given dimensions of the lens and focal length, these 
equations determine n1(z), the index profile of wedge section 42 as a 
function of z. Note that there is no bending of the ray path in the 
gradient region because the propagation direction is parallel to the index 
gradient. 
For small angles, these equations can be solved by expanding the 
trigonometric functions. The index profile is 
EQU n1(z)/n1(0)=1+z*N/D(z), (5) 
EQU where 
EQU N=(y1/z1).sup.2 *(na/n1(0)); (6) 
EQU and 
EQU D(z)-F-L+(na/n1(0))(L+z). (7) 
For large focal length F, the function D(z) is essentially constant, and 
the index profile becomes linear in z, the distance along the axis of the 
gradient segment. Even for finite F, the index varies smoothly. The change 
in index dn needed to produce the focal length F is then 
EQU dn=n1(z1)-n1(0)=y1.sup.2 *na/z*D(z1)!, (8) 
and the focal length can be expressed as 
EQU F=(L+z1)*(1-na/n1(0))+(y1.sup.2 *na/dn-z1.sup.2)/z1. (9) 
A segmented lens with z1=y1, na=1.0, n1(0)=1.5, dn=0.2, well within the 
state of the art, has a focal length of 
EQU F=(L+z1)/3+4*z1. (10) 
For example, there are commercially available standard cylindrical lenses 
with parameters of F=54 mm and y1=12 mm. These values can be matched by a 
segmented axial gradient lens with the above index values and with L=6 mm 
and y1=z1=12 mm. 
FIG. 9 shows the index profile required for a lens of the wedge type shown 
in FIG. 3B. Using the parameters as defined in FIG. 8A, the lens was 
assumed to have a full depth of 2 cm with L=1 cm and z1=1 cm. The range of 
the index of refraction was 1.695 to 1.605 and the homogeneous rear 
segment had an index of refraction of 1.605. The solid curve in FIG. 9 is 
the index profile calculated for a full height of 2 cm, that is, y1=1 cm, 
which yields a focal length F of 10.87 cm. The dashed curve in FIG. 9 is 
the index profile calculated for a reduced height of 0.67 cm, y1=0.335 cm, 
which yields a focal length F of 1 cm; thus F is equal to L, so that the 
image of an object at infinity will occur on the rear surface of the lens. 
In FIG. 8B, a three segment design utilizing two axial gradient wedges 62, 
94 is shown. This design will function as a beam expander/contractor or as 
a beam reshaper depending upon the index of refraction profiles that are 
chosen. The index of refraction profile in the first wedge 62 is denoted 
by n1(z) and in the second wedge 94 by n2(z'). The index of the central 
homogeneous segment 92 is equal to n1(0) which is equal to n2(0). L is the 
distance between the apex of segment 62 and the apex of segment 94. A2 is 
the base angle for segment 94. The equations that determine the path for 
an incident horizontal ray displaced a distance y above the optical axis 
12 which emerges horizontally at a distance y' above the axis are 
EQU n1(z) sin (A1)=n1(0) sin (A1+B) (11) 
EQU n1(0) sin (A2-B)=n2(z') sin (A2) (12) 
EQU tan (B)=(y-y')/L+z+z'! (13) 
EQU z=y*tan (A1)=y*z1/y1 (14) 
EQU z'=y'*tan (A2)=y'*z2/y2. (15) 
A symmetric treatment holds for rays incident below the central optical 
axis 12. For given dimensions of the lens, focal length and functionality, 
these equations determine n1(z) and n2(z'), the index profiles of the two 
wedges. 
For small angles, these equations can be solved by expanding the 
trigonometric functions. The index profiles are 
EQU n1(z)/n1(0)=1+(y1/z1)*B(z) (16) 
EQU n2(z')/n1(0)=1-(y1/z1)*B(z'), (17) 
EQU where 
EQU B=(y-y')/L+z+z'!, (18) 
and the ray deflection angle B can be expressed as a function of z or of z' 
to fully determine n1(z) and n2(z'). 
For the beam expander/contractor, the beam intensity distribution must be 
scaled in the y-direction. This implies that y'=y*y2/y1 which in turn 
requires z'=z*z2/z1, so that 
EQU B(z)=(z/z1)*(y1-y2)/L+z(1+z2/z1)! (19) 
EQU B(z')=(z'/z2)*(y1-y2)/L+z'(1+z1/z2)!. (20) 
For L large compared to (z1+z2), the denominator in B is essentially 
constant, and both axial index profiles become linear in the distance 
along the optical axis of the gradient segment. 
For a beam reshaper, the distribution of intensity in the incident beam 
must be rearranged to the desired final intensity distribution. Thus, the 
exit coordinate y' must be a given function of the entrance coordinate y, 
y'=y'(y). By using this y' value, and the implied relation between z' and 
z in the above equations for the deflection angle B(z), the index 
distributions for the two axial gradient segments for the reshaper are 
completely determined. 
For example, if the incident intensity distribution is i(y) (assumed 
symmetric) and the desired output intensity distribution is flat, the 
requirement of energy conservation forces the relation 
EQU y'-y2*I(y)/I(y1), 
where I(y) is the integral of i(y) from 0 to the point y. This relation 
between y' and y then determines the relation between z' and z, and 
eventually, both of the index profiles. 
If a segmented axial gradient lens is used in an optical system which 
contains other optical elements, the designer can choose the index of 
refraction profile to meet the performance requirements of the total 
system. In short, the index gradient can correct for the aberrations 
induced by several lens elements. For example, the designer can replace an 
ordinary element with a segmented axial gradient lens and choose its index 
profile so that the aberrations are minimized in one transverse direction. 
A second segmented axial gradient lens mounted at right angles to the 
first can minimize the aberrations in the orthogonal transverse direction. 
The principles and designs for individual segmented axial gradient lenses 
discussed above can be applied to the array lenses of the invention. Thus, 
a cylindrical lens functionality of the lenses of FIGS. 2B, 2D, 3B, 3D, 
and 5A-5C can provide the functionality of an array of parallel 
cylindrical lenses or for a narrow beam input an optical multiplexer. The 
point foci lens functionality of the lenses of FIGS. 4B, 4D, 6 and 7A-C 
can provide the functionality of a two dimensional array of point foci 
such as that of a video monitor. 
Array Lenses 
The wedge, plane and cononical nterfaces of the individual segmented axial 
gradient lenses are implemented in the array lenses of the invention with 
sheets of optical material having parallel grooves or an array of cones at 
the interfaces between sheets. FIG. 10 shows a side sectional view of an 
array lens 140 with a first planar sheet 142 having an axial gradient 
index of refraction profile and a second planar sheet 144 having a 
homogeneous index of refraction. The first sheet includes first and second 
sides with the first side forming a front surface 141 of the array lens 
140. The second sheet includes first and second sides with the second side 
forming the rear surface 143 of the array lens. The second side of the 
first sheet 142 and the first side of the second sheet 144 are provided 
with corresponding parallel grooves and ridges which mate with each other. 
Each of the grooves has two planar walls. The grooves of first sheet 142 
each includes a wall 146 which is parallel to the optical axis and a wall 
148 which forms a finite non-normal angle with the optical axis. The 
optical axis is normal to front surface 141. Line 145 is one optical axis 
of lens 140. It should be noted that for the array lens structures of the 
invention, each lens has a plurality of optical axes. In the case of 
lenses having the functionality of a plurality of parallel cylindrical 
lenses, lines normal to the front surface and lying in planes of light 
rays not bent by the lens can be considered optical axes. In the case of 
an array lens which produces an array of foci in an image plane, lines 
normal to the front surface and containing the point foci will be optical 
axes. Those of ordinary skill in the art will understand the positioning 
of the optical axes for various other configurations of the array lenses 
of the invention. The walls of adjacent grooves in sheet 142 meet to form 
ridges 147 and the walls of adjacent grooves in sheet 144 meet to form 
ridges 149. A beam of parallel rays 150 is shown entering the front 
surface 141 of the array lens 140. The rays are bent at the interface 
between the sheets 142 and 144 and at the rear surface 143 of the array 
lens 140 (except for rays along the optical axes where the indices of 
refraction of the two sheets are exactly equal) in the manner as described 
above with reference to FIGS. 2B and 8A. A series of parallel line foci 
152 are formed at an image plane 154 of the lens 140. Thus, the array lens 
140 provides the functionality of a sequence of parallel cylindrical 
lenses. The lens array will also function if planar sheet 142 has a 
homogeneous index of refraction and sheet 144 contains a suitable index of 
refraction profile. 
Referring now to FIG. 11, an array lens 160 is shown in cross section 
having a first axial gradient sheet 162 and a second homogeneous sheet 
164. The array lens 160 functions in a manner similar to the array lens 
140 of FIG. 10 except that both walls of each groove in the two sheets 
form finite non-normal angles with with ray parallel to the optical axes. 
These angles may be selected along with the gradient profile to provide 
the desired focal length. As with lens 140 of FIG. 10, array lens 160 
provides a series of parallel line foci formed at an image plane of the 
lens. The lens array will also function if planar sheet 162 has a 
homogeneous index of refraction and sheet 164 contains a suitable index of 
refraction profile. 
FIG. 12 shows a perspective view of an array lens 170. Lens 170 includes a 
first planar sheet 172, an intermediate planar sheet 174 and a last planar 
sheet 176. Sheets 172 and 176 each has an axial gradient index of 
refraction profile. The parallel grooves and ridges which form the 
interface between sheets 172 and 174 are rotated 90 degrees around a 
central axis 175 relative to the grooves and ridges which form the 
interface between sheets 174 and 176. A beam of parallel rays 180 is shown 
entering the front surface of the array lens 170. The rays are bent at the 
interfaces between the sheets 172 and 174, the sheets 174 and 176, and at 
the rear surface of the array lens 170 in the manner as described above 
with reference to FIG. 6. A two dimensional array of foci 182 are formed 
at an image plane 184 of the lens 170 as seen in FIG. 13. 
Referring now to FIGS. 14A, 14B, and 15, an array lens 190 is provided 
which produces as an output a series of parallel, finite-sized beam 
stripes. Array lens 190 is shown in cross-section in FIG. 14A with a first 
planar sheet 192 having an axial gradient index of refraction profile, an 
intermediate planar sheet 194 having a homogeneous index of refraction, 
and a last planar sheet 196 having an axial gradient index of refraction 
profile. This same array is shown in perspective view in FIG. 14B. First 
sheet 192 includes first and second sides with the first side forming a 
front surface 198 of the array lens 190. Last sheet 196 includes first and 
second sides with the second side forming the rear surface 200 of the 
array lens. The intermediate sheet 194 includes first and second sides 
which interface with the second side of the first sheet and the first side 
of the last sheet, respectively. The interfaces between the adjacent 
sheets are provided with mating grooves and ridges. A beam of parallel 
rays 202 is shown entering the front surface 198 of the array lens 190. 
The rays are bent at the interfaces between the respective sheets 192, 194 
and 196. In this embodiment, the rays are not bent at the rear surface 200 
since they exit the lens normal to the surface. Thus, the rays are bent at 
the interfaces in the manner as described above with reference to FIGS. 5 
A-5C and 8B. In FIG. 14B, an output viewing surface 204 is depicted beyond 
the lens 190. As shown in FIG. 15, a series of parallel beam stripes 206 
are provided as the output of array lens 190 upon the surface 204. 
FIGS. 16 and 17 provide examples of one use of the lenses of the invention. 
FIG. 16 shows an optical multiplexer 210 using the lens structure 
described with reference to FIG. 11. An input port 212 and two output 
ports 214 such as can be achieved with optical fibers or light pipes are 
provided. The optical fibers or light pipes may be joined to the lens 
structure in a manner known to those skilled in the art. The multiplexer 
210 includes a gradient index of refraction segment 216 and a homogeneous 
segment 218. The interface side of segment 216 includes one full groove 
and two half grooves and the interface side of segment 218 includes two 
full grooves. An input beam 220 at port 212 will generally diverge as 
shown and be focused at the interface to two output beams 222 and 224. 
In the multiplexer 230 shown in FIG. 17, a homogeneous, low index of 
refraction wedge section 232 is provided to interface with input port 212. 
This will provide more beam divergence at the input location. 
FIG. 18A shows a perspective view of an array lens sheet 242 that utilizes 
cones as the interface surfaces. As seen in FIG. 18B which is a side view 
of an assembled lens 240 the lens includes first planar sheet 242 and last 
planar sheet 248. Sheet 248 has an axial gradient index of refraction 
profile and a series of conical indentations on its from surface and a 
planar rear surface 247. Sheet 242 has a planar front face 249 and a 
series of cones 246 on its rear face that fit into the indentations in 
sheet 248. The planar surface 244 is in contact with the planar surface of 
sheet 248. A two dimensional array of foci are formed at an image plane to 
the rear of the array lens 240. 
The number of intermediate layers in the array lenses of the invention is 
not limited to one. Nor is the intermediate layer limited to have a 
homogeneous index of refraction. Multiple intermediate sheets may be 
utilized with none, some or all having an axial gradient index of 
refraction profile. The particular configuration may be selected to 
provide the desired magnification and focal length for the lens. 
It should be understood that various alternatives to the embodiments of the 
invention described herein may be employed in practicing the invention. It 
is thus intended that the following claims define the scope of the 
invention and that structures and methods within the scope of these claims 
and their equivalents be covered thereby.