Note: Descriptions are shown in the official language in which they were submitted.
-- 1 --
FOCUSING ELEMENTS AND SYSTEM FOR PRODUCING A
PRESCRIBED ENERGY DISTRIBUTION ALONG AN AXIAL FOCAL ZONE
This invention relates generally to universal focus-
ing systems, and more particularly to axicon-type focusing
elements.
In the area of optical recording and playback appara-
tus, the accurate resolution of data on an optical data record
has proven difficult to obtain. U.S. Patent Nos. 3,501,586
and 4,090,031 to Russell and 4,1~2,209 to Hedlund, et al.
disclose examples of such apparatus.
Typically, optical data is recorded along a track in
the form of a very fine optical pattern, such as close-
ly-spaced microscopic dots. This data is played back by scan-
ning a light beam along the track to modulate the light beam
in accordance with the optical pattern. The modulated beam is
either transmitted or reflected to a light detector which
produces an electrical output signal in accordance with the
modulation of the beam. If all goes well, this signal faith-
fully reproduces the optical pattern to play back the origi-
nally recorded signal. However, numerous factors affect theability to faithfully reproduce the recorded signal.
One important factor is the ability to focus the beam
in the plane of the optical pattern and to maintain focus
during playback. Prior playback systems employ a fixed or
adjustable focal length focusing system including spherical or
other point-focusing lenses. The focal point of the focusing
system is positioned in the plane of the optical pat-tern by a
variety of techniques. Examples of such techniques include
moving either the optical elements or the record or both along
the optical axis.
Initially, the optical elements and record are posi-
tioned manually. Because the optical pattern is typically
very small and dense, positioning must be very precise. The
problem is compounded by the need for accurate angular align-
ment of such elements. Thus, positioning the optical elements
`'i~
~ ~3~.Y33
can be very difficult and time consuming.
Once the optical elements and record are initially
positioned, focusing continues to be a problem during the
operation of the playback system. The op-tical records are
seldom truly planar. The mechanisms supporting and moving the
records also have a certain amount of play in them. In the
Hedlund, et al. apparatus of U.S. Patent No. 4,142,209, the
optical elements are shiFted along the optical axis to scan
different tracks. All of these factors can dynamically change
the relative positions of the optical elements and record,
causing the focal point to deviate from its ideal posi-tion.
As a result, the image of data pattern tends to move in and
out of focus causing errors during playback.
Accordingly, there is a need in optical playback
systems for means for focusing the light beam on optical data
patterns which obviates the need for highly precise position-
ing of the focusing elements and record. Such beam-focusing
means should also be immune to dynamic variations in the rela-
tive position of its elements and the record in an optical
20 pl ayback apparatus.
In the field of optics, researchers have long sought
a universal focus lens. In 1954, J. H. McLeod published a
paper entitled "The Axicon: A New Type of Optical Element" in
the Journal of the Optical Society of America, No. 44, pp.
592-597, indicating the discovery of such a lens. In his
paper, McLeod defined the axicon and described several exam-
ples: a toric lens, a right conical lens and reflector, a
hollow refractive sphere and a hollow reflective cylinder.
McLeod also described possible uses for axicons in telescopes,
microscopes, projectors and autocollimators.
Since 1954, further examples and applications of
axicons have been identified. In an article entitled "Focons
and Foclines as Concentrators of the Radiation of Extended
Objects," Sovie-t Journal of Optical Technology, February 1977,
pp. 66-69, V. K. Baranov describes two families of reflecting
axicons--the parabolic-toroidal focon and the parabolic-cylin-
drical focline--and their ability to concentrate light from an
3~3
extended source to a point. M. Rioux, et al. discussed the
use of axicons in combination with lasers in an article enti-
tled ~'Linear, Annular, and Radial Focusing with Axicons and
Applicaiions to Laser Machining," Applied Optics, Vol. 17, No.
10, ppO 1532-1536, May 15, 1978.
Axicons have also been found to be applicable to
forms of energy other than electromagnetic. Acoustical axi-
cons and their applications are discussed in two articles by
C.~. Burckhardt, et al. entitled "Ultrasound Axicon: A Device
for Focusing Over a Large Depth," Journal of the Acoustical
Society of America, No. 6, 1973, pp. 1628-30 and "Methods For
Increasing The Lateral Resolution of B-Scan," Acoustical
Holography, Vol. 59 1973, pp. 391-413, and in an article by
H. D. Collins entitled "Acoustical Interferometry Using Elec-
tronically Simulated Variable Reference And Multiple Path
Techniques," Acoustical Holography, Vol. 6~ 1975, pp. 597-619.
Certain imaging characteristics of conical axicons
are analyzed in an article entitled "Imaging Properties of a
Conic Axicon," Applied Optics, August 1974, pp. 1762 1764 by
W. R. Edmonds. In an article entitled "Focal Depth of a
Transmitting Axicon," Journal of tne Optical Society of
America, April 19i3, pp. 445-449, J. W. Y. Lit, et al.
examined the axial field distribution of conical and curved
conical axicons for plane uniform incident energy and for
gaussian distribution incident energy. Lit, et al. noted that
the focal zone intensity distribution would vary from point to
point along the axis of a conical axicon.
However~ none of the foregoing references deal with
the problems of employing an axicon in an optical playback
system. The references suggest that something approaching a
universal focusing system can be made using an axicon. How-
ever, they do not recognize the problems that the variation of
intensity of energy focused onto the axis of an axicon can
create in an optical recording and playback system. For
example, with a conical axicon, the variation in focal zone
intensity can be great enough to completely mask the modula-
tion of a light beam scanned across an optical record.
-- 4 --
U.S. Patent ~o. 4,133,600 to Russell, et al. uses a
conical axicon lens to try to alleviate tolerance requirements
in regard to positioning or flatness of the record. However,
the axicon is used only for formation of holographic lens
means, not during recording or playback of data.
Likewise, none of the references suggest an axicon
focusing system or method of making axicons which will elimi-
nate or control this intensity variation. Accordingly, there
remains a need for a focusing system which will provide both
nearly universal focusing and a controllable intensity dis-
tribution within a focal zone for playing back data on an
optical record.
It is, therefore, an object of the invention to focus
a beam of energy on an axial point regardless of changes in
the position of such point along a finite length line.
A second object of the invention is to focus the beam
with a specifiable intensity distribution along such line.
A third object is to focus the beam as aforesaid with
a uniform intensity distribution along such line.
~O Another object of the invention is to focus energy as
aforesaid from a beam having a gaussian radial intensity dis-
tribution.
A further object is to focus energy from a collimated
beam, such as that produced by a laser, into an axial focal
zone with a specifiable energy distribution along a portion of
such zone.
A still further object of the invention is to provide
a method for making an energy-focusing element, such as a lens
or reflector, usable either alone or with other elements, to
focus a beam of energy into an axial focal zone with a pre-
scribed intensity distribution.
An additional object of the invention is to provide a
focusing element having a refractive boundary operable to
focus a beam of energy to a finite length line with a specifi-
able intensity distribution along such line.
Another object is to provide a family of focusing
elements as aforesaid in which the refractive boundary is
3~
adjustable to accommodate incident energy beams having differ-
ing axially symmetrical in-tensity distributions to produce a
specifiable focal zone intensity distribution.
A further object is to provide a method of making
Focusing elements having the foregoing characteristics.
A specific object of the invention is to provide for
a family of axicon-like optical-focusing elements having a
nonconical, aspherical refractive surface.
Another specific object is to provide for a family of
axicon-like optical lenses having an aspherically curved sur-
face and an opposite concave or convex surface.
A further specific object is to provide an optical
playback system with means for focusing light on data in an
optical record so as to produce a substantially uniformly
modulated light beam regardless of errors in the relative
position of the record and optical elements of the system
along its optical axis.
To satisfy the foregoing objects, we have invented
both a family of nonconical curved, axicon-type focusing ele-
ments and a method for making such elements. In the contextof this invention, the terms "focusing element~" and "axicon"
are used interchangeably to refer to any kind of energy-focus-
ing means. They include both reflective elements, or mirrors,
and transmissive elements, or lenses. Such elements can have
discrete boundaries or surfaces, or they can be formed of
continuous focusing media. They can also include lens and
reflectors for focusing acoustical energy such as ultrasonic
waves, optical lenses and mirrors and antennae for focusing
microwaves.
One feature of the invention is an axially symmetri-
cal, nonconical focusing means configured to focus an axially
symmetrical incident energy beam to a line, or axial focal
zone, having a prescribed energy distribution between two
different points. Another feature of the invention is a
method of making such a focusing means.
Yet another feature is that the focusing means can be
specified for an incident energy beam having a particular
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radial intensity distribution. This feature enables the lens
to be matched to both the characteristics of the incident
energy beam and the desired intensity distribution along the
focal zone.
The foregoing and other objects, features and advan-
tages of the invention will become more apparent from the
following detailed description of a preferred embodiment of
the invention, which proceeds with reference to the accompany-
ing drawings.
Fig. 1 is a schematic diagram of one embodiment of an
optical playback apparatus for a reflective-type optical data
record, incorporating a focusing system according to the pres-
ent invention.
Fig. 2 is an enlarged, vertical cross-sectional view
of a concave-convex axicon lens such as that of Fig. 1 showing
the focusing of incident energy from a point source to a focal
zone.
Fig. 3 is a line diagram of the convex surface of the
lens of Fig. 2, the unbroken rays indicating refraction of the
incident energy and the dashed rays indicating reflection of
such energy when the surface is used as a concave mirror.
Fig. 4 is a front elevational view of the surface of
Fig. 3-
Figs. 5a, 5b and 5c are examples of incident energy
intensity distributions from various sources, Fig. 5a showinga uniform distribution and Figs. 5b and 5c showing truncated
Gaussian distributions.
Fig. 6 is an enlarged vertical cross-sectional view
of a portion of the scanner and record of Fig. 1 illustrating
the focal zone produced by the lens of Fig. 2.
Fig. 7 is a preferred focal zone intensity distribu-
tion produced by a focal system in accordance with the inven
tion.
Fig. 8 is another example of a prescribed focal zone
intensity which a focusing system can be made to produce in
accordance with the invention.
Fig. 9 is a vertical cross-sectional view of a
-
3hl~3
double-convex lens according to the invention.
Referring to Fig. 1, an optical playback apparatus
for playing back data recorded on an optical record 10 com-
prises a light source 12, a light detector means 14, a focus-
in~ means 16 for Focusing light emitted from the light source,and scanning means 18 for scanning the focused beam across
record 10.
The light source 12 is preferably a laser which pro-
duces coherent light in a narrow frequency band. However, any
essentially point source of light can be used which produces
an axially symmetrical light beam 20.
Beam 20 is directed toward focusing means 16. The
focusing means preferably includes a primary point-focusing
lens 22, an apertured light mask, commonly referred to as a
"pinhole" element 24, and a "tipping" plate 26. The foregoing
elements are aligned along the optical axis of beam 20. Lens
22 focuses beam 20 through the pinhole of element 24 to shape
the light spot for imaging onto the optical record 10. The
tipping plate is positioned in the path of the shaped light
beam to laterally displace the light beam through small angles
by refraction. The tipping plate is pivoted by a galvano-
meter-type motor 28 in response to a tracking signal applied
to its input 30 to cause the light beam to stay on the data
track being scanned. The details of construction and opera-
tion of the foregoing elements are discussed in U.S. PatentNos. 3,891,794 and 4,090,031 to Russell and, hence, need not
be further discussed herein.
The focusing means 16 also includes a beam-splitter
mirror 32 and an axicon lens 34 positioned to receive beam 20
from the tipping plate. The beam splitter is positioned`
between lens 34 and the tipping plate so as to transmit light
from lens 34 in one direction and to reflect a return light
beam 36 to light detector 14. Alternatively, lens 34 can be
positioned between the pinhole element and the tipping plate.
It is also possible to eliminate the pinhole element and lens
22 altogether. Other positions for lens 34 are also possible.
-- 8 --
From the focusing means, beam 20 passes to the scan-
ning means 18. The scanning means comprises a rotating scan-
ner wheel 38 mounted for rotation on a shaft 40. The shaFt is
driven at a constant speed by electrical motor 42 coupled
thereto. The axis of rotation of shaft 40 is parallel to the
optical axis of lens 34. A rotating distributor mirror member
44 is mounted on the end of shaft 40 adjacent lens 34. The
distributor member 44 has an outer surface in the form of a
five-sided pyramid, having five mirrors provided thereon which
correspond to five objective lenses 46 carried on the scanner
wheel near its outer periphery~ Associated with each of these
lenses is a mirror 48 positioned to reflect light from one of
the sides of the distributor member into one of the lenses
46. The distributor member rotates with the scanner wheel at
the same angular velocity as the lenses 46 and mirrors 48.
Accordingly, each mirror of the distributor member is always
optically aligned with its associat~d objective lens. The
distributor mirror member is offset slightly from the optical
axis of lens 34 so that it distributes light beam 20 to the
20 objective lenses 46 one at a time as such lenses rotate across
record 10 to play back the optical data recorded thereon.
Absent lens 34, lenses 22 and 46 focus light beam 20
to a small spot along the optical axis 50 of lens 46 in a
selected focal plane intermediate points 52, 54 as described
in UOS. Patent No. 4~090,031. In prior playback apparatus,
painstaking efFort was required to insure that this focal
plane coincided with the plane of the optical data on record
10, indicated by reference numeral 56.
The function of lens 34 is to extend the depth of
focus of the light beam along axis 50 so as to focus every-
where in a focal zone between points 52 and 54. Consequently,
so long as the data pattern 56 is located along the optical
axis 50 within this focal zone, it will be in focus.
As beam 20 scans across record 16, it is transmitted
35 through the record or reflected back in accordance with the
optical pattern on the record. The reflected light forms
return beam 36 which is modulated in accordance with the
g
optical data pattern on the record. This beam passes in the
reverse direction through lens ~6, mirror 48 and the mirrors
of distributor member 44 along the same path traversed by beam
20. The beam splitter then reflects beam 36 to detector means
14. The detector means is a photodetector which is sensitive
to the frequency band of return beam 36 to produce an output
signa`l on output 58 which corresponds to the modulation of the
return light beam.
In order to provide a suitable amplitude of modulated
light beam 36 to detector means 14, it is preferable for beam
36 to vary only in response to modulation by the data pattern
on the record. That is, the average amplitude of beam 36
should not vary as a function of the position of the data
pattern along the focal zone. This requires a focusing means
capable of focusing light to a specified, preferably uniform,
axial intensity distribution along at least a portion of the
focal zone between points 52 and 54, as shown in Figs. 6 and
7. A specific structure for a family of lenses, including
lens 34, capable of producing such a distribution is the sub-
ject of the next subsection.
An example of lens 34 is shown in Fig. 2. Lens 34 is
constructed of a refractive medium such as optical glass hav-
ing an index of refraction n. The surrounding medium has an
index of refraction m.
Lens 34 is a species of axicon. In its more general
definition, the term "axicon" means an optical or other
energy-focusing element which is a figure of revolution and
which images energy from a point source 60 along its axis 62
as a line in a focal zone 64 extending between two separate
end points 66, 6~3. The distinguishing feature of the optical
elements of the present invention is that they are designed to
provide a prescribed intensity distribution along the focal
zone. Such elements have a nonconical, refractive boundary,
such as surface 70 in lens 34, which is aspherically-curved so
as to focus light to such a distribution. In the case of a
lens, the element has a second refractive boundary or surface
72 which is also a figure of revolution, but which is designed
,
3~
-- 10 --
to focus light to a point. Surface 72 is typically spherical
and can be concave, as shown in Fig. 2, planar (not shown)~ or
convex, as shown in Fig. 9.
In the past, lenses having aspherically-curved sur-
faces were difficult to generate. In recent years, opticalmachining technology has reached a stage of development which
perrnits generation of complex surfaces of revolution. ~achine
tools of this type are disclosed in U.S. Patent ~o. 4,083,272
to Miller, in an article by D. M. Miller, et al. entitled
"Precision Machining of Optics," Proceedings of the Society of
Photo-Optical Instrumentation Engineers, Vol. 159, pp. 32~41,
August 1978 and in an article by T. T. 5aito entitled "Diamond
Turning of Optics," Optical Engineering, ~ol. 17, Nov.-Dec.,
1978, teachings of which are incorporated herein by refer-
ence. The following paragraphs illustrate how a lens isdesigned in accordance with the invention and specified in
terms usable by the apparatus of U.S. Patent No. 4,083,272, to
produce surface 70.
Referring to Fig. 2, surface 72 is assumed to be a
concave spherical surface whose radius of curvature equals its
distance from source 60. As a result, the energy emanating
from source 60 and incident upon surface 72, indicated by a
paraxial ray 74 and two pairs of nonaxial rays 76,78, is un-
refracted as it passes through surface 72 into lens 34. Thus,
the entire refractive power of lens 34 is concentrated in
surface 70.
Accordingly, the analysis and design of surface 70
can be simplified by reference to Fig. 3 in which surface 72
is omitted. The region to the left of surface 70 in Fig. 3
has an index of refraction n and the region to the right of
surface 70 has an index of refraction m. For convenience, the
optical axis 62 is referred to as the x-axis. Points radially
offset from the x-axis are referred to as having a position p,
which is radially symmetrical about the optical axis. Accord-
ingly, points on an axial cross section of surface 70 can bedefined by variable coordinates (x, p), as is further explain-
ed hereinafter.
~ ~ ~3~
11 -
Referring to Figs. 2 and 4, energy emanating from
point source 60 is focused by different radial portions of
surface 70 into different axial portions of focal zone 64.
Paraxial rays 74 passing through a small circle 80 on surface
70 of radius dp about axis 62 are focused to point 66. The
off-axis rays 76, 78 passing -through annular rings 82, 84 of
radial width dp on surface 70 are focused further away from
such surface. Ray 78 passes through ring 84 at the periphery
of surface 70 at a distance R from axis 62. Accordingly, it
is focused to the end point 68 of focal zone 64. Ring 82 at a
distance p, which less than R, from axis 62, focuses ray 76 to
point 67. The position of point 67 along the x-axis is a
variable function U(p) depending on the distance p of ring 82
from axis 62. The positions of points 66 and 68 can be con-
sidered constants: U(p=o) or simply U(o), and U(p=R) or simply
U(R), respectively.
As mentioned above, a conical or other conventional
axicon (not shown), produces a nonuniform output intensity
distribution of energy focused along its focal zone. Refer
ring to Fig. 4, if the incident energy has a uniform intensity
distribution, as shown in Fig. Sa, less light is focused to a
point at the end of the cone by circle 80 than is focused by
rings 82 or 84 to points more distant from the cGne. Thus,
the output intensity distribution of a conical axicon
increases along its focal zone for incident light of a uniform
intensity distribution.
Lens 34 is designed to produce a uniform or other
prescribed intensity distribution between two separate points
U1 and U2 in focal zor,e 64. Such points must fall between
end points 66 and 68, as shown in Fig. 2 and typically are
selected to coincide with the end points9 as shown in Fig. 6.
When lens 34 is incorporated in the focusing means of Fig. 1,
the energy of beam 20 is focused to a line between points 52,
54 so as to uniformly illuminate the data pattern on the
record so `long as the pattern is between such points. Points
U1 and U2 are chosen far enough apart to allow for rela-
tively loose tolerances in the position of the data pattern,
3~
- 12 -
yet close enough together to avoid design extremes, for exam-
ple, 8 mm. apart.
In designing ~lens 34 in accordance with the inven-
tion, it is assumed that the light source irradiates the sur-
face 70 with light represented by an axially-symmetrical
intensity distribution function, f(p). Examples of such dis-
tribution functions are shown in Figs. 5a, 5b and 5c.
Referring to Fig. 4, the quantity of energy dE pass-
ing through ring 82 is the product of the area of the ring and
the incident energy intensity, that is:
(1)
dE = 2~pf(p)dp.
This amount of energy is focused onto axis 62 in a
zone of length dU, for example, at point 67 in Fig. 2. For a
generalized axial intensity distribution, such as is shown in
Fig. 8,
(2)
dE
dU K(U)
Distribution function K(U) can be specified for the
entire focal zone from U(o) -to U(R) or for a portion thereof,
for eY~ample, between points U1 and U2. For a uniform
axial energy distribution, K(U) is a constant K. Therefore9
(3)
dU = KdE = 2~Kp f(p) dp,
To define U(p), the foregoing expression is inte-
grated over the radius of lens 34, yielding
- 13 -
(4)
U(p) = U(o) + 2~K ~ P f (p)dp
At this stage, it is necessary to specify the input
intensity distribution of light source 12. For a 1aser light
source
(5)
IO 2
f(P) = Io e~kP .
Referring to Figs. 5b and 5c, the light intensity produced by
the laser has a truncated Gaussian distribution. At radius R
from its axis, this distribution has an intensity IR.
Therefore,
(6)
k = - 1~ ln (IR/Io)
Having specified the characteristics of the incident
light source and the desired intensity distribution in the
focal zone, the integration of equation (4) is completed,
yielding
~5
(7)
U(p) = U(o) + ~KI [l_e-kp2]
k
The foregoing equation specifies the focal zone of a
focusing element in terms of the characteristics of a selected
incident light source, the desired focal zone intensity dis-
tribution and the radius p of surface 70. This relationship
is employed in a lens equation -to define the shape of surface
70, as described below.
Fig. 3 illustrates Fermat's Principle, which is
described in Jenkins, et al., Fundamentals of Optics,
McGraw-Hill Pub., 1957, pp. 7-10. This principle specifies
3~
- 14 --
the relationship of the optical path lengths of rays 74 and
76. Paraxial ray 74 passes from point source 60 at a position
a along the x-axis to axial focal point 67 at position b
through an axial point 80 on surface 70 at position c. Ray 76
passes from point 60 to point 67 through an off-axis point 82
on surface 70 at axial and radial coordinates (x, p). Fer-
mat's Principle states that the optical path lengths of the
two rays 74, 76 are equal, yielding the following lens equation
(8)
n ~f (x-a)2 + p2 ~ m ~,/ (x-b)2 + p2 = n(c-a) + m(b-c)
where n and m are the indices of refraction of the media on
the left and right sides, respectively, of surface 70.
This relationship can be used, as is, in apparatus
such as that of U.S. Patent No. 4,083,27~q to generate lens and
mirror surfaces capable of focusing light to a focal point.
However, the present invention requires a surface 70 capable
of focusing incident light onto an extended focal zone 64 with
a prescribed intensity distribution, as shown in Fig. 2.
Accordingly, the lens equation is modified as follows to
specify position b as a focal zone rather than a focal point:
qc (9)
n J (x-a)2 + p2 + m J (x-U(p))2+p2 = n(c-a) + m(U(p)-c)
This equation is used as described in the following example to
produce lens 34.
The following parameters are specified for this
example only. Lens 34 has an index of refraction n = 1.488.
The index of refraction of air is m - 1. Lens surface 72 is
concavely spherical about a radius of 63.2 mm. The thickness
of lens 34 along axis 62 is 9.0 mm. By specifying c = 0, the
~'.r33~3~3
value of a = -72.2. As a result, equation (9) becomes
(10)
1.488 ~ (x+72.2)2 + p2 ~ ~(x-U(p))2+p2 = 1.488 x 72.2 + U(p)
At the maximum lens radius R = 1.5 mm, the incident
light intensity IR ~ .5 Io yielding k = 0.308. The axial
points along the focal zone 64 between which it is desired to
have a uniform intensity distribution can be freely chosen,
for example, U(o) = 40 mm and U(R) = 48 mrn. As a result
(11)
U(p) = 40 + 16 [1_e-0.308p2]
Using the foregoing parameters~ lens equation (10) is
then solved by iterative techniques for various values of p to
obtain a se-t of coordinates (x, p) which define surface 70~
To numerically control a lens grinding apparatus, such as that
of U.S. Patent No. 4,083,272, values of x would be computed
from equation (10) for incremental values of p in the
sub-micron range.
Alternatively, one can compute a small number of
values of x for larger increments of p, for example, 0.2 mm
increments, as shown in first and third columns of Table 1
below. The second column lists values of U(p) corresponding
to each increment of p.
TABLE 1
p (mm) U(p) (mm) x XC xc - x
0.0 40.00000 0.0
0.2 40.19591 -0.00187 -0.001720.00015
0.4 ~0.76937 -0.00740 -0.006900.00050
0.6 41.67926 -0.01644 -0.015530.00091
0.8 42.86253 -0.02880 -0.027620.00118
1.0 44.24135 -0.04426 -0.043180.00108
1.2 45.73163 -0.06264 -0.~62240.000~0
1.4 47.25127 -().08385 -0.08479-0.00094
1.6 48.72744 -0.10782 -0.11087-0.00305
- 16 -
Ihese values of x and p are used to generate a fourth
or higher order polynomial characterizing surface 70 in a form
which lens makers are presently accustomed to using. The
first step is to curve fit a spherical surface to the fore-
going values. The standard spherical lens equation
(12)
n-m _ n + m
r (c-a) ~
provides a first approximation. Using the exemplary values
for n, m, a and setting b = U(p=0), the spherical surface has
a radius of curvature r ~ 10.7mm. Comparison of the coordi-
nates of such a surface with the coordinates of surface 70listed in Table 1 yields a monotonically increasing error
function. Therefore, a somewhat larger radius r better fits
surface 70. For a central value of p, for example p = 0.8,
r ~ 11.6 mm. At this radius, the error at the axis of sur-
face 70 is approximately equal to the negative of the error
at the periphery of surface 70, that is, at R = 1.5 mm.
The next step is to compute values XC along the
x-axis for the values of p in column 1 of Table 1 by solving
the equation for a circle of radius r centered on the x-axis
at x = -11.6 mm.
(13)
XC = -11.6 + 113~.56-p2
These values are listed in the fourth column of lable
1. The last column contains values of the difference ~x
between corresponding values of the third and fourth columns.
This difference is then fitted to a fourth power
equation of the form
(14)
~x = xc-x = a1 p + a2 p2 + a3 p2 + a4 p4
3~3
- 17 _
Using the data of Table 1, surface 70 can then be
specified by
(15)
x = xc -~x = -11.6 + ~134.56-p2 f 0.000610p -
0.00691~ p2 ~ 0.005951p3 -0.00070 p 4
This specification of surface 70 yields a precision
of about 50 nanometers, which is much less than the wavelength
of visible light and approaches the accuracy of existing
lens-turning apparatusO
Equation (9) can also be used to specify the shape o-f
an axicon reflector surface in accordance with invention for
focusing light along dashed lines 76' to axial position b' by
specifying the indices of refraction n = 1 and m = -1.
Using the parameters of Example 1, the surface of
- such a reflector would be defined by equation (11) and equa-
tion (16) below.
(16)
~ (x~72.2)2 ~ p2 + l (x-U(p))2 f p2 = 72.2 - U(p)
It should be noted that, due to reflection, the sign of U(p)
is negative.
As mentioned above, axicon lenses in which surface 72
is not a concave ~phere can also be designed in accordance
with the invention. One example is a double convex lens 100,
shown in Fig. 9. Lens 100 has a convex axicon lens surface
102 and a convex point-focusing lens surface 104 in lieu of
surface 72. Following is a specification of surface 102 and
104 in accordance with the invention.
Surface 104 transmits to surface 102 an irnage of
pOillt source 60 which appears to be located at a position
other than its actual location a. For example, surface 104
can be designed to bend rays 106 from source 60 inwardly
~3~
toward axis 62. Tracing these rays -toward the source along
dashed 1ines 108, source 60 appears at a position a' more
distant from the lens. The shape of surface 104 can be
specified to obtain any position a' by subs-tituting a' for b
in equation (8) and, additionally, specifying the values for
the indices of refraction m and n and positions a of the
source 60 and c' of the apex of surface 104 along the x-axis
yielding,
(17)
~(x a)2 + p2 + m ~(x-a')2+ p2 = n (c'-a) + m (a'-c')
To specify surface 102, the position a' is determined
from equation (17) and substituted for position a in equation
(9), yielding
(18)
~ 2 2 + ~ (x U(p))2 ~ p2 = n(c-a~) + m(U(p)-c)
Equations (17) and (18) can be used in the manner
described in Example 1 to control a lens-turning apparatus to
generate surfaces 104 and 102, respectively. However, the
procedure can be further simplified by specifying surface 104
as one which transmits to surface 102 an image of source 60
positioned an inFinite distance from lens 34, as indicated by
dashed lines 90 in Fig. 9.
With surface 104 so specified, a' ~ - ~D and equa-
tion (18) simplifies to
(19)
nx + m ~(x-U(p))2 + p2 = (n-m)c+mU(p)
This equation is readily solved for values of x and p
to control a lens-turning machine in making surface 102.
~3~3
- 19 -
Although specific examples of optical elements have
been described, the present invention includes many other
energy-focusing elements designed to focus incident energy
into an axial focal zone with a prescribed intensity distribu-
tion. For example, equation (19) can also be used to specifyan acoustical axicon lens for an ultrasonic transducer.
Energy-focusing systems can, within the scope of the inven-
tion, be arranged differently from the system described here-
in. The invention can also be embodied in different forms of
optical playback appara-tus.
Having illustrated and described a preferred embodi-
ment of the invention and several examples thereof, it should
be apparent to those skilled in the art that the invention may
be modified in arrangement and detail. ~e claim as our inven-
tion all such modifications as come within the true spirit andscope of the following claims.
