U.S. patent number RE38,839 [Application Number 10/192,104] was granted by the patent office on 2005-10-18 for methods and devices to design and fabricate surfaces on contact lenses and on corneal tissue that correct the eye's optical aberrations.
Invention is credited to Peter C. Magnante.
United States Patent |
RE38,839 |
Magnante |
October 18, 2005 |
Methods and devices to design and fabricate surfaces on contact
lenses and on corneal tissue that correct the eye's optical
aberrations
Abstract
Methods and devices are described that are needed to design and
fabricate modified surfaces on contact lenses or on corneal tissue
that correct the eye's optical aberrations beyond defocus and
astigmatism. The invention provides the means for: 1) measuring the
eye's optical aberrations either with or without a contact lens in
place on the cornea, 2) performing a mathematical analysis on the
eye's optical aberrations in order to design a modified surface
shape for the original contact lens or cornea that will correct the
optical aberrations, 3) fabricating the aberration-correcting
surface on a contact lens by diamond point turning, three
dimensional contour cutting, laser ablation, thermal molding,
photolithography, thin film deposition, or surface chemistry
alteration, and 4) fabricating the aberration-correcting surface on
a cornea by laser ablation.
Inventors: |
Magnante; Peter C. (West
Brookfield, MA) |
Family
ID: |
23577804 |
Appl.
No.: |
10/192,104 |
Filed: |
July 10, 2002 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
Issue Date |
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Reissue of: |
399022 |
Sep 20, 1999 |
06086204 |
Jul 11, 2000 |
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Current U.S.
Class: |
351/212;
219/121.75 |
Current CPC
Class: |
A61B
3/1015 (20130101) |
Current International
Class: |
A61B
3/14 (20060101); A61B 3/103 (20060101); A61B
3/15 (20060101); A61B 003/10 () |
Field of
Search: |
;351/205,211,212,219,221,246,160R,176,177 ;606/4,5
;219/121.6,121.61,121.69,121.75 |
References Cited
[Referenced By]
U.S. Patent Documents
Other References
"Aberration-Correcting Contact Lenses to Improve Human Vision",
Peter C. Magnante, Nancy J. Coletta, and Bruce D. Moore, Brookfield
Optical Systems, West Brookfield, MA 01585, New England College of
Optometry, Boston, MA 02115. .
"Plastic Optics" by Richard F. Weeks, Sep. 1975, Optic News, pp.
5-11. .
International Search Report, Application No.
PCT/USOO/22466..
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Primary Examiner: Manuel; George
Attorney, Agent or Firm: Conte; Robert F.I. Barnes &
Thornburg
Claims
What I claim as my invention is:
1. A method for correcting the optical aberrations beyond defocus
and astigmatism of an eye fitted with an original contact lens
having a known anterior surface shape by providing a modified or
new contact lens which has its anterior surface reshaped from said
original contact lens's anterior surface, comprising the steps of:
a) measuring said optical aberrations of an eye fitted with an
original contact lens, b) performing a mathematical analysis of
said eye's optical aberrations when fitted with original contact
lens to determine said modified anterior contact lens surface
shape, and c) fabricating said modified anterior contact lens
surface by methods that remove, add or compress material or alter
the surface chemistry.
2. A method as claimed in claim 1 wherein said measuring of the
eye's optical aberrations comprises the sub-steps of: i) optically
projecting the image of a small point of incoherent light onto the
macular region of the eye's retina, ii) optically conveying the
image of the eye's pupil, through which light scattered back from
the macular region emerges, onto a microlens array, iii) optically
conveying the multiple spot images formed by said microlens array
onto the image plane of a photo-electronic imaging device, iv)
transforming by means of the photo-electronic imaging device the
multiple spot images formed by said microlens array to an
electronic signal which represents the images, v) conveying said
electronic signal to a computer for data processing, vi) processing
first the electronic signal with said computer in order to obtain
the coordinate locations of the centroids of said multiple spot
images formed by said microlens array, and vii) processing next
said coordinate locations with said computer in order to obtain the
slopes of optical rays emerging from the subject's pupil at said
coordinate locations.
3. A method as claimed in claim 1 wherein said mathematical
analysis comprises the sub-steps of: i) determining mathematically
the normal vectors of said original contact lens's anterior
surface, ii) determining mathematically the directional derivatives
of said modified or new contact lens's anterior surface using data
of said normal vectors of original contact lens's anterior surface
and data of said eye's optical aberrations, and iii) fitting
mathematically by the method of least squares said directional
derivatives to the corresponding directional derivatives of a
polynomial expression that represents said modified or new contact
lens's anterior surface.
4. A method as claimed in claim 1 wherein said step of fabricating
said modified or new contact lens's anterior surface is chosen from
the group of methods comprising diamond point machining, laser
ablation, thermal molding, photo-lithographic etching, thin film
deposition, and surface chemistry alteration.
5. A method for correcting the optical aberrations beyond defocus
and astigmatism of an eye with an original anterior corneal surface
of known shape by providing a modified anterior corneal surface
shape, comprising the steps of: a) measuring said eye's optical
aberrations, b) performing a mathematical analysis of said eye's
optical aberrations to determine said modified anterior corneal
surface shape, c) fabricating said modified anterior corneal
surface by laser ablation.
6. A method as claimed in claim 5 wherein said measuring of the
eye's optical aberrations comprises the sub-steps of: i) optically
projecting the image of a small point of incoherent light onto the
macular region of the eye's retina, ii) optically conveying the
image of the eye's pupil, through which light scattered back from
the macular region emerges, onto a microlens array, iii) optically
conveying the multiple spot images formed by said microlens array
onto the image plane of a photo-electronic imaging device, iv)
transforming by means of the photo-electronic imaging device the
multiple spot images formed by said microlens array to an
electronic signal which represents the images, v) conveying said
electronic signal to a computer for data processmg, vi) processing
first the electronic signal with said computer in order to obtain
the coordinate locations of the centroids of said multiple spot
images formed by said microlens array, and vii) processing next
said coordinate locations with said computer in order to obtain the
slopes of optical rays emerging from the subject's pupil at said
coordinate locations.
7. A method as claimed in claim 5 wherein said mathematical
analysis comprises the sub-steps of: i) determining mathematically
the normal vectors of said original anterior corneal surface, ii)
determining mathematically the directional derivatives of said
modified anterior corneal surface using data of said normal vectors
of original anterior corneal surface and data of said eye's optical
aberrations, and iii) fitting mathematically by the method of least
squares said directional derivatives to the corresponding
directional derivatives of a polynomial expression that represents
said modified anterior corneal surface.
8. An ophthalmic device for measuring the eye's optical aberrations
either with or without a contact lens in place on the cornea,
including; a) an optical projection system for imaging a small
point of light onto the macular region of the eye's retina with an
improvement provided by use of an incoherent light source chosen
from the group comprising laser diodes operated below threshold,
light emitting diodes, arc and plasma sources, and incandescent
filament lamps, b) an optical image acquisition system for
conveying the image of the eye's pupil, through which light
scattered back from the macular region emerges, onto a microlens
array, c) a microlens array to form multiple spot images onto the
image plane of a photo-electronic imaging device, d) a
photo-electronic imaging device for transforming said multiple spot
images formed by said microlens array to an electronic signal which
represents the images, e) a computer for processing the electronic
signal in order, first, to obtain the coordinate locations of the
centroids of said multiple spot images formed by said microlens
array and, second, to obtain the slopes of optical rays emerging
from the subject's pupil at said coordinate locations, and f) an
optical alignment system allowing the entering beam to be
accurately centered with respect to the subject's pupil.
9. An ophthalmic device as claimed in claim 8 wherein said optical
projection system includes an optical isolator consisting of a
quarter-wave plate and polarizer.
10. An ophthalmic device as claimed in claim 8 wherein said optical
projection system includes a field stop placed at a location that
is optically conjugate to the eye's retina.
11. An ophthalmic device as claimed in claim 8 wherein said optical
projection system includes both an optical isolator consisting of a
quarter-wave plate and polarizer, and a field stop placed at a
location that is optically conjugate to the eye's retina.
12. An ophthalmic device as claimed in claim 8 wherein said
photo-electronic imaging device is chosen from the group comprising
vidicons, charge-coupled devices, and charge-injection devices.
13. A device for thermally forming surfaces on thermoplastic
contact lens blanks that correct eyes' optical aberrations beyond
defocus and astigmatism consisting of a die with an adjustable
surface shape (either continuous or discontinuous) formed by
computer-controlled electromechanical actuators or
electromechanical fingers which are known in the field of adaptive
optics.
14. A lathe device for machining surfaces on contact lens blanks
that correct eyes' optical aberrations beyond defocus and
astigmatism consisting of a rotating spindle onto which a contact
lens blank is fastened, translation slides for precisely
positioning a diamond point cutting tool with respect to the
surface of the contact lens blank, and a programmed computer that
controls the movement of the translation slides synchronously with
the rotational location of the spindle.
15. A contour cutting device for machining surfaces on contact lens
blanks that correct eyes'optical aberrations beyond defocus and
astigmatism consisting of a means for supporting and holding
stationary a contact lens blank, translation slides for precisely
positioning in three dimensions a diamond point cutting tool with
respect to the surface of the contact lens blank, and a programmed
computer that controls the movement of the translation
slides..Iadd.
16. A method for correcting optical aberrations beyond defocus and
astigmatism of an eye comprising: a) fitting the eye with a first
contact lens having a known anterior surface shape that is
corrected for at least focus, b) measuring the optical aberrations
of the eye fitted with the first contact lens, c) performing a
mathematical analysis of the eye's optical aberrations when fitted
with the first contact lens to determine a modified anterior
contact lens surface shape, and d) fabricating a second contact
lens having the modified anterior contact lens
surface..Iaddend..Iadd.
17. A method as claimed in claim 16 performing the mathematical
analysis to at least the eye's 4.sup.th order optical aberrations
of the eye when fitted with the first contact lens to determine the
second contact lens surface shape, and the anterior surface of the
second contact lens to have the modified anterior surface that
corrects the eye to at least the 4.sup.th order
aberration..Iaddend..Iadd.
18. A method as claimed in claim 17 wherein the measuring of the
eye's optical aberrations comprises the sub-steps of: i) optically
projecting the image of a small point of incoherent light onto the
macular region of the eye's retina, ii) optically conveying the
image of the eye's pupil, through which light scattered back from
the macular region emerges, onto a microlens array, iii) optically
conveying the multiple spot images formed by the microlens array
onto the image plane of a photo-electronic imaging device, iv)
transforming by means of the photo-electronic imaging device the
multiple spot images formed by the microlens array to an electronic
signal which represents the images, v) conveying the electronic
signal to a computer for data processing, vi) processing first the
electronic signal with the computer in order to obtain the
coordinate locations of the centroids of the multiple spot images
formed by the microlens array, and vii) processing next the
coordinate locations with the computer in order to obtain the
slopes of optical rays emerging from the subject's pupil at the
coordinate locations..Iaddend..Iadd.
19. A method as claimed in claim 17 wherein the mathematical
analysis comprises the sub-steps of: i) determining mathematically
the normal vectors of the first contact lens's anterior surface,
ii) determining mathematically the directional derivatives of the
second contact lens's anterior surface using data of the normal
vectors of the first contact lens's surface and data of the eye's
optical aberrations, and iii) fitting mathematically by the method
of least squares the directional derivatives to the corresponding
directional derivatives of a polynomial expression that represents
the second contact lens's anterior surface..Iaddend..Iadd.
20. A method as claimed in claim 1 wherein the original contact
lens's anterior surface contour function z (x,y) and the eye's
optical aberrations, represented by optical rays emerging from the
pupil given by vector function B (x,y), are used to find the
surface contour function z' (x,y) of the modified or new contact
lens by the following mathematical procedures: a) z'(x,y) is
approximated by the sum of a series of linearly independent terms
in x & y with each term labeled by an index j wherein each
term, a.sub.j.multidot.g.sub.j (x,y), consists of an unknown
constant coefficient a.sub.j and a known function g.sub.j (x,y) as
shown in following Equation (1)
##EQU18## b) following Equation (2) is obtained by taking vector
gradients of both sides of Equation (1) where grad z' (x,y) is a
two-dimensional vector having components ##EQU19## c) the normal
vectors of the original contact lens's anterior surface, given by
vector function N(x,y), are found by taking the vector gradient of
z(x,y) and normalizing it to unity as shown in the following
Equations (3A) to (3D) ##EQU20##
MAG.ident.[1+(.delta.z(x,y)/.delta.x).sup.2
+(.delta.z(x,y)/.delta.y).sup.2 ].sup.1/2 (3D) d) the rays incident
on the original contact lens coming from within the eye are given
by vector function A(x,y) which is found by applying Snell's law of
refraction at the air/lens interface which relates A(x,y) to known
vector function B(x,y) representing the emerging rays, and vector
function N(x,y) given by Equations (3A)-(3D), e) the normal vectors
of the modified or new contact len's anterior surface, given by the
vector function N', are found from the following Equation (4) where
vector function B' is represented by unit vectors pointed along the
positive z-axis, and n is the lens's refractive index ##EQU21## f)
the directional derivatives of the modified or new contact lens's
anterior surface are obtained from the following Equation (5) using
the components of vector function N'=[N'.sub.x, N'.sub.y, N'.sub.z
] found from Equation (4) ##EQU22## g) apply the method of least
squares to minimize the square the difference between the component
values of grad z'(x,y) found from Equation (5) and the component
values of grad z'(x,y) given by the approximation series, Equa.
(2), in order to obtain matrix Equation (6)
where ##EQU23## h) obtain the inverse of matrix M, and then find
the a-coefficients from matrix Equation (8)
21. A methdo as claimed in claim 19 wherein the first Contact
lens's anterior surface contour function z(x,y) and the eye's
optical aberrations, represented by optical rays emerging from the
pupil given by vector function B(x,y), are used to find the surface
contour function z'(x-y) of the second contact lens by the
following mathematical procedures: a) z' (x,y) is approximated by
the sum of a series of linearly independent terms in x & y with
each term labeled by an index j wherein each term,
a.sub.j.multidot.g.sub.j (x,y), consists of an unknown constant
coefficient a.sub.j and a known function g.sub.j (x,y) as shown in
following Equation (1) ##EQU24## b) following Equation (2) is
obtained by taking vector gradients of both sides of Equation (1)
where grad z' (x,y) is a two-dimensional vector having components
##EQU25## c) the normal vectors of the first contact lens's
anterior surface, given by vector function N(x,y), are found by
taking the vector gradient of z(x,y) and normalizing it to unity as
shown in the following Equations (3A) to (3D) ##EQU26##
MAG.ident.[1+(.delta.z(x,y)/.delta.x).sup.2
+(.delta.z(x,y)/.delta.y).sup.2 ].sup.1/2 (3D) d) the rays incident
on the first contact lens coming from within the eye are given by
vector function A(x,y) which is found by applying Snell's law of
refraction at the air/lens interface which relates A(x,y) to known
vector function B(x,y) representing the emerging rays, and vector
function N(x,y) given by Equations (3A)-(3B), e) the normal vectors
of the second contact lens's anterior surface, given by the vector
function N', are found from the following Equation (4) where vector
function B' is represented by unit vectors pointing along the
positive z-axis, and n is the lens's refractive index ##EQU27## f)
the directional derivatives of the second contact lens's anterior
surface are obtained from the following Equation (5) using the
components of vector function N'=[N'.sub.x,N'.sub.y,N'.sub.z ]
found from Equation (4) ##EQU28## g) apply the method of at least
squares to minimize the square the difference between the component
values of grad z' (x,y) found from Equation (5) and the component
values of grad z' (x,y) given by the approximation series, Equa.
(2), in order to obtain matrix Equation (6)
where ##EQU29## h) obtain the inverse of matrix M, and then find
the a-coefficient from matrix Equation (8)
22. A method as claimed in claim 5 wherein the original cornea's
anterior surface contour function z(x,y) and the eye's optical
aberrations, represented by optical rays emerging from the pupil
given by vector function B(x,y), are used to find the surface
contour function z'(x,y) of the modified cornea by the following
mathematical procedures: a) z' (x,y) is approximated by the sum of
a series of linearly independent terms in x & y with each term
labeled by an index j wherein each term, a.sub.j.multidot.g.sub.j
(x,y), consists of an unknown constant coefficient a.sub.j and a
known function g.sub.j (x,y) as shown in following Equation (1)
##EQU30## b) following Equation (2) is obtained by taking vector
gradients of both sides of Equation (1) where grad z' (x,y) is a
two-dimensional vector having components ##EQU31## c) the normal
vectors of the original cornea's anterior surface, given by vector
function N(x,y), are found by taking the vector gradient of z(x,y)
and normalizing it to unity as shown in the following Equations
(3A) to (3D) ##EQU32## MAG.ident.[1+(.delta.z(x,y)/.delta.x).sup.2
+(.delta.z(x,y)/.delta.y).sup.2 ].sup.1/2 (30) d) the rays incident
on the original cornea coming from within the eye are given by
vector function A(x,y) which is found by applying Snell's law of
reflection at the air/cornea interface which relates A(x,y) to
known vector function B(x,y) representing the emerging rays, and
vector function N(x,y) given by Equations (3A)-(3B), e) the normal
vectors of the modified cornea's anterior surface, given by the
vector function N', are found from the following Equation (4) where
vector function B' is represented by unit vectors pointing along
the positive z-axis, and n is the cornea's refractive index
##EQU33## f) the directional derivatives of the modified cornea's
anterior surface are obtained from the following Equation (5) using
the components of vector function N'=[N'.sub.x, N'.sub.y, N'.sub.z
] found from Equation (4) ##EQU34## g) apply the method of least
squares to minimize the square the difference between the component
values of grad z'(x,y) found the Equation (5) and the component
values of grad z' (x,y) given by the approximation series, Equa.
(2), in order to obtain matrix Equation (6)
where ##EQU35## h) obtain the inverse of matrix M, and then find
the a-coefficients from matrix Equation (8)
23. A contact lens comprising an anterior surface which is
fabricated to correct the optical aberration to at least 4.sup.th
order of a person's eye..Iaddend..Iadd.
24. The contact lens of claim 23 wherein the measuring of the eye's
optical aberrations is by steps of: (i) optically projecting the
image of a small point of incoherent light onto the macular region
of the eye's retina, (ii) optically conveying the image of the
eye's pupil, through which light scattered back from the macular
region emerges onto a microlens array, (iii) optically conveying
the multiple spot images formed by the microlens array onto the
image plane of a photo-electronic imaging device, (iv) transforming
by means of the photo-electronic imaging device the multiple spot
images formed by the microlens array to an electronic signal which
represents the images; (v) conveying the electronic signal to a
computer for data processing, (vi) processing first the electronic
signal with the computer in order to obtain the coordinate
locations of the centroids of the multiple spot images formed by
the microlens array, and (vii) processing next the coordinate
locations with the computer in order to obtain the slopes of
optical rays emerging from the subject's pupil at the coordinate
locations..Iaddend..Iadd.
25. The contact lens of claim 23 wherein the mathematical analysis
comprises the sub-steps of: (a) determining mathematically the
normal vectors of a first contact lens anterior surface; (b)
determining mathematically the directional derivatives of the a
second contact lens anterior surface using data of the normal
vectors of the first contact lens anterior surface and data of the
eye's optical aberrations; and (c) fitting mathematically by the
method of least squares the directional derivatives to the
corresponding directional derivatives of a polynomial expression
that represents the second contact lens anterior
surface..Iaddend..Iadd.
26. The contact lens of claim 23 wherein the first contact lens
anterior surface contour function z(x,y) and the eye's optical
aberrations, represented by optical rays emerging from the pupil
given by vector function B(x,y), are used to find the surface
contour function z'(x,y) of the second contact lens by the
following mathematical procedures: (a) z'(x,y) is approximated by
the sum of a series of linearly independent terms in x & y with
each term labeled by an index j wherein each term,
a.sub.j.multidot.g.sub.j (x,y), consists of an unknown constant
coefficient a.sub.j and a known function g.sub.j (x,y) as shown in
following Equation (1) ##EQU36## b) following Equation (2) is
obtained by taking vector gradients of both sides of Equation (1)
where grad z' (x,y) is a two-dimensional vector having components
[.delta.z'(x,y)/.delta.x, .delta.z'(x,y)/.delta.y] ##EQU37## c) the
normal vectors of the first contact lens's anterior surface, given
by vector function N (x,y), are found by taking the vector gradient
of z(x,y) and normalizing it to unity as shown in the following
Equations (3A) to (3D) ##EQU38##
MAG.ident.[1+(.delta.z(x,y)/.delta.x).sup.2
+(.delta.z(x,y)/.delta.y).sup.2 ].sup.1/2 (3D) d) the rays incident
on the first contact lens coming from within the eye are given by
vector function A(x,y) which is found by applying Snell's law of
refraction at the air/lens interface which relates A(x,y) to known
vector function B(x,y) representing the emerging rays, and vector
function N(x,y) given by Equations (3A)-(3B), e) the normal vectors
of the second contact lens's anterior surface, given by the vector
function N', are found from the following Equation (4) where vector
function B' is represented by unit vectors pointing along the
positive z-axis, and n is the lens's (or cornea's) refractive index
##EQU39## f) the directional derivatives of the second contact
lens's anterior surface are obtained from the following Equation
(5) using the components of vector function N'=[N'.sub.x, N'.sub.y,
N'.sub.z ] found from Equation (4) ##EQU40## g) apply the method of
least squares to minimize the square the difference between the
component values of grad z' (x,y) found from Equation (5) and the
component values of grad z'(x,y) given by the approximation series,
Equa. (2), in order to obtain matrix Equation (6)
where ##EQU41## h) obtain the inverse of matrix M, and then find
the a-coefficients from matrix Equation (8)
Description
1. BACKGROUND OF THE INVENTION
1.1 Measurements of the Eye's Aberrations
There are several objective optical techniques that have been used
to measure the wavefront aberrations of the eye. The aberroscope,
which is disclosed by Walsh et al. in the Journal of the Optical
Society of America A, Vol.1, pp. 987-992 (1984), projects a nearly
collimated beam into the eye which is spatially modulated near the
pupil by a regular grid pattern. This beam images onto the retina
as a small bright disk which is modulated by the dark lines of the
grid pattern. Since the eye's pupillary aberrations distort the
retinal image of the grid pattern, measurements of the distortions
on the retina reveal the pupillary aberrations.
The spatially resolved refractometer, which is disclosed by Webb et
al. in Applied Optics, Vol. 31, pp. 3678-3686 (1992), projects a
small diameter collimated beam through the eye's pupil. Instead of
being spatially modulated by a physical grid as with the
aberroscope, the spatially resolved refractometer's beam is
raster-scanned across the entire pupil. A sequence of retinal
images of the focused light is recorded with each image associated
with a particular location at the pupil. A mapping of the relative
locations of these retinal images reveals the aberrations across
the pupil.
Analyzers of retinal point-spread functions have been disclosed by
Artal et al. in the Journal of the Optical Society of America A,
Vol. 5, pp. 1201-1206 (1988). Analyzers of retinal line-spread
functions have been disclosed by Magnante et al. in Vision Science
and Its Applications, Technical Digest Series (Optical Society of
America, Washington, D.C.), pp. 76-79 (1997). When used to measure
the wavefront aberrations of the eye, these spread function
analyzers project a small diameter circular beam into the eye at
the center of the pupil. This beam focuses onto the retina as a
tiny source of light. The light from this tiny retinal source
scatters back through the dilated pupil. A small circular aperture
(approximately 1 mm diameter) in the imaging section of the
analyzer is located conjugate to the pupil plane. This aperture may
be translated up/down or side/side to sample specific regions in
the pupil plane where wavefront aberration measurements are sought.
An imaging lens focuses the light through the small aperture onto
the imaging plane of a camera. Measurements of the relative
locations of the focal spots for the various locations of the small
aperture characterize the pupillary wavefront aberrations.
The Hartmann-Shack wavefront sensor for ordinary lens or mirror
testing was disclosed originally by Shack et al. in the Journal of
the Optical Society of America, Vol. 61, p. 656 (1971). This type
of wavefront sensor was adapted to measure the wavefront
aberrations of the eye by Liang et al., Journal of the Optical
Society of America A, Vol. 11, pp. 1949-1957 (1994). The
Hartmann-Shack wavefront sensor is similar to point-spread (or
line-spread) function analyzers in that: 1) it projects a fine
point of light onto the retina through a small diameter pupil of
the eye, and 2) the light which is scattered back from the retina
through the eye's pupil is imaged onto a camera with a lens that is
conjugate to the eye's pupil. However, instead of using a single
lens with a moveable small aperture to image the retinal image onto
the camera, the Hartmann-Shack wavefront sensor utilizes a regular
two-dimensional array of small lenses (commonly called a microlens
array) which is optically conjugate to the eye's pupil to focus the
back scattered light from the retinal image onto the camera.
Typical diameters of individual microlenses range from 0.1 to 1.0
millimeter. With the Hartmann-Shack wavefront sensor, instead of
having a single spot of light corresponding to a single aperture
imaged by the camera, there is an array of focused spots imaged by
the camera . . . one spot for each lens in the microlens array.
Furthermore, each imaged spot of light corresponds to a specific
location at the eye's pupil. Measurements of the locations of the
array of imaged spots are used to quantify the pupillary
aberrations.
Measurements of the wavefront aberrations of the eye to a high
degree of precision using an improved Hartmann-Shack wavefront
sensor are described in 1998 U.S. Pat. No. 5,777,719 to Williams
and Liang. What is described in U.S. Pat. No. 5,777,719 improves
upon what was described previously by Liang et al. in the Journal
of the Optical Society of America A, Vol. 11, pp. 1949-1957 (1994).
Device improvements described in the Williams and Liang 1998 Patent
include: 1) a wavefront correcting deformable mirror, 2) a method
to feedback signals to the deformable mirror to correct the
wavefront aberrations, and 3) a polarizer used with a polarizing
beamsplitter to reduce unwanted stray light from impinging on the
recording camera.
Although the precision of the resulting wavefront aberration
measurements cited by Williams and Liang is impressive, the
implementation of a deformable mirror and a feedback loop is very
costly and is not necessary for achieving the purposes of my
invention.
Furthermore, the polarizer with polarizing beamsplitter cited in
the Williams and Liang patent are not necessary for achieving the
purposes of my invention, and those devices are replaced in my
invention with a single device called an optical isolator
(consisting of a polarizer fused to a quarter-wave plate). The
optical isolator achieves the same purpose as the pair of
polarizing devices described by Williams and Liang, namely reducing
unwanted stray light.
Finally, a laser is cited as the preferred illumination source in
the Williams and Liang patent. However, a conventional laser is
improved upon in my invention through the use of a diode laser
operated below threshold. Such a light source is not as coherent as
a standard laser operating above threshold, Images formed with such
a non-coherent source are less granular (having less "speckle")
than those formed by coherent sources. This improvement results in
less noisy granularity in the microlens images and, thereby,
improves the accuracy of the image processing which depends on
precisely locating the microlens images.
1.2 Analysis of Hartmann-Shack Wavefront Sensor Data to
Characterize the Eye's Optical Aberrations
The essential data provided by a Hartmann-Shack wavefront sensor
modified to measure the human eye are the directions of the optical
rays emerging through the eye's pupil. The method of deriving a
mathematical expression for the wavefront from this directional ray
information is described by Liang et al. in the Journal of the
Optical Society of America A, Vol. 11, pp. 1949-1957 (1994). It is
also the method cited in 1998 U.S. Pat. No. 5,777,719 to Williams
and Liang. First, the wavefront is expressed as a series of Zernike
polynomials with each term weighted initially by an unknown
coefficient. Zernike polynomials are described in Appendix 2 of
"Optical Shop Testing" by D. Malacara (John Wiley and Sons, New
York, 1978). Next, partial derivatives (in x & y) are then
calculated from the Zernike series expansion. Then, these partial
derivative expressions respectively are set equal to the measured
wavefront slopes in the x and y directions obtained from the
wavefront sensor measurements. Finally, the method of least-squares
fitting of polynomial series to the experimental wavefront slope
data is employed which results in a matrix expression which, when
solved, yields the coefficients of the Zernike polynomials.
Consequently, the wavefront, expressed by the Zernike polynomial
series, is completely and numerically determined numerically at all
points in the pupil plane. The least-squares fitting method is
discussed in Chapter 9, Section 11 of "Mathematics of Physics and
Modern Engineering" by Sokolnikoff and Redheffer (McGraw-Hill, New
York, 1958).
Although the above described methods to calculate the aberrated
wavefront of the eye are cited in the Williams and Liang patent, it
is significant to note that there is not any description in their
patent of how to design an aberration-correcting contact lens or
corneal surface from the aberrated wavefront data. These details
for designing an aberration-correcting contact lens or corneal
surface are not obvious, and require a number of complex
mathematical steps. These mathematical details for designing
aberration correcting surfaces on contact lenses or on the cornea
itself are described fully in my invention.
Furthermore, Williams and Liang demonstrate that the eye's
aberrations can be corrected by properly modifying the surface of a
reflecting mirror. However, they do not demonstrate or provide any
description of how aberration-correcting surfaces can be designed
on refractive surfaces such as those on contact lenses or on the
cornea itself. My invention gives a detailed mathematical
description of how to design such refracting optical surfaces that
correct the eye's aberrations.
1.3 Fabrication of Conventional Contact Lenses
Conventional contact lenses with spherical or toroidal surface
contours are made routinely using a method called single point
diamond turning which utilizes very precise vibration-free lathes.
The contact lens blank rotates on a spindle while a diamond point
tool, moving along a precise path, cuts the desired surface
contour. The end result is a surface which does not need additional
polishing, and exhibits excellent optical qualities in both figure
accuracy and surface finish. Figure accuracy over the lens surface
is better than one wavelength of light. Surface finish, which is
reported as rms surface roughness, is better than 1 micro-inch.
Machines of this type and their use are described by Plummer et al.
in the Proceedings of the 8th International Precision Engineering
Seminar (American Society of Precision Engineering, pp. 24-29,
1995).
1.4 Corneal Tissue Ablation to Correct Vision
With the advent of the excimer laser, the means are available for
refractive surgeons to flatten and reshape the surface of the
cornea in order to improve vision. The excimer laser selectively
removes microscopic layers of corneal tissue allowing light rays to
focus more sharply on the retina. In the procedure known as
photorefractive keratectomy (PRK), the laser ablates tissue on the
surface of the cornea. In the procedure known as laser in-situ
keratomileusis (LASIK), the surgeon first creates a flap on the
cornea and then uses the laser to reshape tissue below the corneal
surface. Layers of tissue as thin as 0.25 microns can be
ablated.
With current laser procedures, it is possible only to correct
relatively coarse or low order aberrations of the eye, namely high
levels of nearsightedness, and moderate amounts of farsightedness
and astigmatism. With the analytical methods of my invention, which
take into account the corneal shape, and both the low order and
higher order aberrations of the eye, a modified corneal shape is
found which allows all rays from external point objects to focus
sharply on the retina. By the means offered by my invention,
refractive surgery procedures to improve vision will be improved
greatly.
2. SUMMARY OF INVENTION
Conventional spectacles and contact lenses are able to correct the
visual acuity of most people to 20/20 or better. For these
individuals, the most significant refractive errors are those
caused by the so-called lowest order optical aberrations, namely
defocus, astigmatism and prism. However, there are many people with
normal retinal function and clear ocular media who cannot be
refracted to 20/20 acuity with conventional ophthalmic lenses
because their corneal surfaces are extraordinarily irregular. In
this group are patients with severe irregular astigmatism,
keratoconus, corneal dystrophies, post penetrating keratoplasty,
scarring from ulcerative keratitis, corneal trauma with and without
surgical repair, and sub-optimal outcome following refractive
surgery. The eyes of these people have abnormal amounts of higher
order or irregular optical aberrations. An objective of the
invention is to improve the vision of these patients. A further
objective is to provide the best vision possible to individuals
with ordinary near and farsightedness and astigmatism. To achieve
these objectives, methods and devices are described that are used
to design and fabricate modified surfaces on contact lenses or on
corneal tissue that correct the eye's optical aberrations beyond
defocus and astigmatism.
The objectives of the invention are accomplished by measuring the
wavefront aberrations of a subject's eye (either with or without a
contact lens on the cornea) using a device that projects a small
point of light on the retina near the macula, re-images the light
scattered back from the retina that emerges from the pupil onto a
microlens array, and records the focal spots formed from this light
when it is imaged by a microlens array on the image plane of an
electronic camera. The image formed on the camera is conveyed to a
computer. The computer utilizes methods to determine the
coordinates of the focal spots and then to calculate the wavefront
slopes of rays emerging from the subject's eye.
The objectives of the invention are further accomplished by
mathematical methods which analyze the wavefront slope data as well
as the shape of the subject's original contact lens or corneal
surface in order to design a modified surface shape for the
original contact lens or cornea that corrects the aberrations. The
steps in this mathematical method are: 1) determining the normal
vectors to the original contact lens or corneal surface, 2) from
these normal vectors and the wavefront slope data, determine the
partial derivatives of the surface for the modified contact lens or
corneal surface that corrects the aberrations, and 3) fitting these
partial derivatives of the aberration-correcting surface with the
corresponding partial derivatives of a polynomial expression that
best represents the aberration-correcting surface. From these
methods, a mathematical expression for the aberration-correcting
surface is obtained.
The objectives of the invention are accomplished finally by
providing devices and methods to fabricate the modified
aberration-correcting surfaces designed by the mathematical methods
described. For contact lenses, these fabrication devices and
methods include those of diamond point micro-machining, laser
ablation, thermal molding, photolithography and etching, thin film
deposition, and surface chemistry alteration. For corneal tissue
resurfacing, these fabrication devices and methods are those
associated with laser ablation as used with photorefractive
keratectomy (PRK) and laser in-situ keratomileusis (LASIK).
The invention may be more clearly understood by reference to the
following detailed description of the invention, the appended
claims, and the attached drawings.
3. DESCRIPTION OF DRAWINGS
!
4. DETAILED DESCRIPTION OF THE INVENTION
4.1 Wavefront Sensor for Measuring the Eye's Optical
Aberrations
A schematic drawing of a wavefront sensor which has been modified
to measure the eye's optical aberrations is shown in FIG. 1. The
design and operating principles of the subassemblies of the
wavefront sensor are explained in detail below.
4.1.1 Projection System
In order to reduce bothersome "speckle" from coherence effects from
conventional laser sources, non-coherent optical sources are
preferred. Thus, source 1 can be anyone of the following: laser
diode operating below threshold, light emitting diode, arc source,
or incandescent filament lamp. The source beam is deflected by fold
mirror 2, and focused by microscope objective lens 3 onto a small
pinhole aperture 4 having a diameter typically in the range from 5
to 15 microns. Another lens 5 collimates the beam which next passes
through polarizer 6 and then through aperture stop 7. Typically
this stop restricts the beam diameter to about 2 mm or less.
Following the stop is an electronic shutter 8 used to control the
light exposure to the patient during measurement to about 1/10 sec.
Beamsplitter 9 deviates the collimated beam by 90 degrees. The beam
then passes through optical isolator 10 which consists of a
quarter-wave plate and polarizer. Lens 11 forms a focused point
image at the center of field stop 12 which the subject views
through focusing eyepiece 13. The subject's eye 15 then images the
light to a point spot on the retina 16. The field stop 12 and the
optical isolator 10 both serve the important function of blocking
bothersome corneal specular reflections and instrument lens
reflections from reaching photo-electronic imaging device 20 such
as a vidicon camera, a charge-coupled device or CCD camera, or a
charge-injection device camera.
4.1.2 Camera System
Since the retina acts as a diffuse reflector, some of the light
from the retinal point image 16 is reflected back out of the eye 15
through the pupil and cornea 14. The beam emerging from the eye has
its polarization randomized due to passage through the eye's
birefringent cornea 14 and lens 17 as well due to scattering by the
diffuse retina 16. Passing now in reverse direction through lens
13, field stop 12, lens 11, and optical isolator 10, the beam,
which is now aberrated by the eye's optics, is incident from the
right side onto beamsplitter 9 which transmits about half its
intensity straight through to relay lens pair 18. Collectively,
lenses 13, 11, and relay lens pair 18 serve to re-image subject's
pupil 14 onto the plane of the microlens array 19 with unit
magnification. In this way the aberrant wavefront emerging from the
subject's pupil 14 is mapped exactly onto the microlens array 19.
As shown in FIG. 2, each tiny lens 23 of the array images a portion
of the aberrated wavefront 26 onto its focal plane 21 at or near
its axis 25. The regular array of microlenses 19 produces a
corresponding array of focal spots 24. Deviations of focal spots
from the respective microlens axes 25 manifest the wavefront slope
error over the entire surface of the microlens array (and,
correspondingly, the subject's pupil). The input image plane of CCD
camera 21 coincides with the focal plane 21 of the microlens array.
Photo-electronic imaging device 20 interfaces with a computer
equipped with a "frame-grabber" board (not shown) controlled by
appropriate software. An image of the array of focal spots 24
formed by the microlens array 19 appears "live" on the computer's
monitor, and is ready for "capture" by the computer when a
measurement is taken.
4.1.2.1 Wavefront Slope Measurement
The nature of the wavefront slope measurement is explained now in
greater detail. If a perfect plane wave is incident normally onto a
perfect lens which has a small aperture near its surface, the rays
passing through the aperture will be focused by the lens to the
lens's focal point located on the lens's axis. Regardless of the
location of the small aperture with respect to the lens surface,
the imaged point will be at the same location. On the other hand,
suppose the wavefront is imperfect (i.e. individual rays randomly
directed and not parallel to the perfect lens's optical axis). The
rays going through the small aperture now form an image at the
lens's focal plane that is displaced from the lens's focal point.
The displacement of the centroid of the imaged spot (between the
perfect and imperfect wave measurements) divided by the distance
between the lens and its focal plane (i.e. the focal length of the
lens) equals the angular slope (measured in radians) of the
wavefront at the location of the small aperture. Repeating this
type of measurement for many locations of the small aperture over
the lens surface fully characterizes the wavefront slope errors at
the various measurement locations on the lens surface. In a
wavefront sensor such as shown in FIG. 1, the moveable small
aperture with single large diffraction-limited lens is replaced
with an array of identical microlenses 19 where each one samples
the wavefront at a particular location. Details of the microlens
array and imaging camera used in the wavefront sensor are shown in
FIG. 2. At the focal plane 21 of the microlens array 19 is the
imaging surface of a photo-electronic imaging device 20 which
records the locations of the focal spots 24 for all the microlenses
in the array. The displacement of each focal spot 24 from the
optical axis of its associated microlens 25 divided by the focal
length of the microlens array equals the slope of the wavefront at
the microlens's location. The locations of the optical axes of the
individual microlenses are determined by a calibration procedure
that involves doing a measurement when an unaberrated wavefront 27
(i.e. uniform plane wave) is incident perpendicularly onto
microlens array 19. Such an unaberrated wavefront is obtained by
replacing the human eye 15 shown in FIG. 1 with a
diffraction-limited lens and imaging screen placed at the
diffraction-limited lens's focal plane.
4.1.3 Pupil Alignment System
In FIG. 1, field stop 12 is a small hole bored in the direction of
the instrument's main optical axis through a mirrored planar
substrate oriented at 45 degrees. The field stop is at the focal
plane of the subject's eyepiece 13. Another eyepiece 22 (called the
examiner's eyepiece) is oriented at 90 degrees to the instrument's
main optical axis so that the examiner can view the subject's pupil
14 by the means provided by lens 13 and the mirrored planar
substrate of field stop 12. Optionally, a small video camera can be
attached to examiner's eyepiece 22 so that a video image of the
subject's pupil 14 can be viewed on a monitor. By either of these
means, the examiner can accurately position the subject's eye 15 so
that the entering beam is accurately centered with respect to the
subject's pupil 14. The positioning of the subject's eye with
respect to the instrument beam is controlled by a mechanism (not
shown) consisting of an x-y-z translation stage that moves a chin
and head rest used by the subject.
4.1.4 Data Acquisition and Processing
The subject is asked to look through eyepiece 13 at the point of
light formed within field stop 12, while the examiner adjusts the
location of subject's eye 15 so that the beam passes through the
center of the pupil 14. Prior to taking a measurement, the examiner
focuses eyepiece 13 trying to achieve the brightest and
best-focused image of the array of focal spots seen on the computer
monitor. When the instrument is aligned with respect to the
patient's eye and a best-focus image is obtained, the operator
presses a key which commands the computer to acquire an image of
the array of spots. During a measurement session, as many as ten
successive images may be acquired for subsequent averaging to
improve the signal/noise ratio of the data. The image analysis
program carries out the following steps: 1) subtracts "background"
light from the image, 2) determines the x & y coordinates (in
pixels) of the centroid for each of the focal spots, 3) subtracts
the x & y pixel coordinate values from a corresponding set of
reference values (obtained from a calibration with a
diffraction-limited reference lens), 4) multiplies the difference
values (in pixel units) by a calibration factor which gives for
each location in the pupil the components in the x and y directions
of the wavefront slope error measured in radians. The components of
the wavefront slope error, labeled Bx and By in the following
sections, are the essential measurement data of the wavefront
sensor.
4.2 Design of Aberration-Correcting Lens from Analysis of Wavefront
Sensor Data
A purpose of the invention is to design modifications to an
initially known lens surface, described by z(x,y), which will
correct the eye's optical aberrations measured with wavefront
sensors through that surface. In this section, the mathematical
equations needed for this task, which leads to a new lens surface
described by z'(x,y), are derived. The equations also are applied
in an illustrative example. The mathematical formalism in this
section is divided into the following parts: 1) description of
original optical surface, 2) obtaining the directional derivatives
of z'(x,y) from the wavefront sensor data, 3) obtaining a
polynomial expansion representing z'(x,y) using the method of least
squares, 4) illustrative example leading to z'(x,y), and 5)
demonstration that z'(x,y) corrects the original aberrations. The
following is a guide to the mathematical symbols (whether primed or
not): a) x & y (and X & Y) are coordinates, b) n, R, Brms,
a.sub.j & b.sub.j are scalars, c) z, .delta.z/.delta.x,
.delta.z/.delta.y, MAG, .alpha., .beta., .lambda., g.sub.j, ERROR
and CUT are functions of x & y, d) N, T, A, B, grad g.sub.j and
grad z are three-dimensional vector functions of x & y, e) a
& b are generalized vectors, and f) M and M.sup.- are
generalized square matrices.
4.2.1 Original Optical Surface
By "original optical surface" is meant the anterior surface of
either a contact lens placed on the cornea or, in the absence of a
contact lens, the cornea itself. The index of refraction associated
with the surface's denser side is n. The original optical surface
is represented by z(x,y) which is the distance of the surface from
various points in the x-y plane of the pupil. The unit vector
perpendicular to the optical surface (called the normal vector N)
has components in the x, y and z directions given by N=(Nx,Ny,Nz)
where: ##EQU1##
The x & y partial derivatives of the function describing the
surface also can be expressed in terms of the components of the
surface normal by rearranging the terms of Equa. 1. ##EQU2##
4.2.2 Obtaining the Partial Derivatives of the Surface Function
which Describes the Aberration-Correcting Optical Surface
The light rays, which emanate from the retinal "point source"
formed by the wavefront sensor's projection system, emerge from the
eye at the original optical surface which is described by z(x,y).
Rays striking the surface from the denser side are described as
A-vectors, and rays leaving the surface into air are described as
B-vectors. Both A and B are vectors of unit length. Refer now to
FIG. 3 to express the A and B vectors in terms of their components
along N (surface normal) and T. Note that T is a unit vector
tangent to the optical surface at the point of conjunction of (and
coplanar with) rays A and B.
Next, substitute the expressions for A and B into
n.multidot.A-B.
The second term vanishes due to Snell's Law of Refraction which
is:
From FIG. 3 and the definition of a vector cross product:
The upright pair of lines in Equa. 8 indicates the magnitude of the
vector enclosed by the pair.
Combining the results above, find the following equations:
##EQU3##
where by definition
Note that the B-vectors are known quantities representing the
wavefront slopes as measured by the wavefront sensor. Using Equas.
9 through 12, the known quantities (B, N and n) determine the
unknown quantities (.beta., .alpha., .lambda. and A).
Next, solutions for the A-vectors are used to find N', the unit
normal to the aberration-correcting optical surface. This new
surface causes the emerging B'-vectors to lie parallel to the
z-axis. In vector form, B'=(0,0,1). The expression for N' is
obtained by rearranging Equa. 11 to conform with this new
situation. ##EQU4##
Finally the following partial derivatives of the new
aberration-correcting optical surface, described by z'(x,y), are
obtained in terms of the components of unit vector N'=(Nx',Ny',Nz')
by using Equa. 3 (only adapted to the new surface parameters):
##EQU5##
4.2.3 Obtaining a Polynomial Expression to Represent z'(x,y)
The next problem is to find the new optical surface z'(x,y) from
the partial derivatives expressed by Equa. 15 and 16, which are
determined at discrete coordinate locations in the pupil plane,
[x.sub.k,y.sub.k ], where the wavefront sensor data are obtained.
Begin by expressing z'(x,y) as a polynomial consisting of linearly
independent terms, g,(x,y). ##EQU6##
The terms, g.sub.j (x,y) , can be almost any mathematical
functions; however, they generally are restricted to products of
powers of x and y, such as x.sup.3.multidot.y.sup.2, or sums and
differences of such products. There is no necessity that the
functions be orthogonal over the plane of the pupil as are, for
example, Zernike polynomials over a circular domain. The
coefficients, a.sub.j, are constants which are determined by the
methods derived below. Define "grad", the gradient, as a
mathematical operator which transforms a scalar function, f(x,y) to
a vector with components in the x and y directions that are,
respectively, the partial derivatives of the function with respect
to x and y.
Applying this operator to both sides of Equa. 17, find ##EQU7##
To find the a-coefficients which provide the "best fit" to the data
(i.e. the grad z' data expressed by Equa. 15 & 16), define the
"ERROR" function as the sum of the squares of the differences
between the grad z'(x,y) data from Equas. 15 & 16 (i.e. the
left side of Equa. 19) and the estimated value of grad z'(x,y)
(i.e. the right side of Equa. 19) for all the measurement points,
designated by the index "k", in the x-y pupil plane. ##EQU8##
For brevity sake, the notation for the variables (x,y) has been
suppressed when writing the functions, g.sub.j (x,y) and z'(x,y),
in Equa. 20. This abbreviated notation also is used in the
equations which follow.
The "method of least squares" determines the a-coefficients by
minimizing the ERROR function. This is done by differentiating
ERROR with respect to each element a.sub.j and setting each
resulting equation to zero: .delta.(ERROR)/ .delta.a.sub.j.ident.0.
There are as many equations resulting from this process as there
are terms in the expansion for z'(x,y) shown in Equa. 17. The
resulting system of linear equations is written: ##EQU9##
Note that the sums over the index "k" imply a summation over all
the (x.sub.k,y.sub.k) coordinates in the pupil plane for the
several g-functions and the grad z' values. Also, the products
indicated in Equa. 21 are vector scalar products or, so-called,
"dot products".
Defining matrix, M, and vector, b, in Equa. 22 & 23 below,
Equa. 21 appears in the much simpler form which is shown by either
Equa. 24 or Equa. 25. Note that M is a symmetric square matrix.
##EQU10##
Thus, finding the a-coefficients is equivalent to solving the
system of linear equations shown in Equa. 24 where matrix, M, and
vector, b, are known quantities. ##EQU11##
Using the conventions of standard matrix algebra, Equa. 24 may be
written:
M.multidot.a.ident.b (25)
The solution for vector, a, follows by using standard methods of
linear algebra to obtain the inverse of matrix M which is
designated by M.sup.-1. Thus, the solution for vector, a, is shown
as Equa. 26.
The set of a-coefficients found from Equa. 26, using values of M
(Equa. 22) and b (Equa. 23), is the final result. With the
a-coefficients determined, the explicit "best fit" polynomial
expansion for z'(x,y), shown by Equa. 17, is determined.
4.2.4 Illustrative Example
Consider, as an example, when the original optical surface is an
acrylic contact lens with an index of refraction n=1.49, and with
an anterior contour in the shape of a paraboloid described by
##EQU12##
The radius of curvature at the apex is 7.8 mm which is the constant
R. All linear dimensions in this example, such as those described
by x, y and z, are understood to be in millimeters.
The partial derivatives of z(x,y) are readily obtained. When these
values are substituted into Equas. 1 and 2, the values for the
components of the normal vector N as functions of x & y are
readily obtained. ##EQU13##
In this example, measurements with the wavefront sensor are spaced
1 mm apart in the x and y directions within the domain of a
circular pupil having an 8 mm diameter. There are 49 measurement
sites. These data, which are the components in x & y of the
emerging B-rays (i.e. Bx & By), are given by the two matrices
shown below. The x-direction is to the right, and the y-direction
is upward. Note that the values of Bx & By are multiplied by
1000 which makes the resulting directional units in milliradians.
##EQU14##
Bx & By represent the x & y directions of the rays emerging
from the contact lens. The X and Y coordinates of the focal spots
resulting from these rays, when imaged by the 1 mm spaced microlens
array onto the wavefront sensor's measurement plane, are shifted
somewhat from the intersection of the regular 1 mm spaced grid
lines. The X & Y coordinate shifts are equal to the product of
the focal length of the microlens array and the Bx and By values.
In this example f=200 mm. The centroids of the focal spots of the
wavefront sensor pattern are shown in FIG. 4.
A measure of the degree of collimation of all the B-rays is its
root-mean-square value, labeled Brms. It is found by taking the
square root of the average of all 49 values of Bx.sup.2 +By.sup.2
as shown in Equas. 28 & 29. The value is Brms=0.00148
radians.
Continuing with the methods to find the correcting surface,
substitute the known numerical data (i.e. N summarized by Equa. 27,
Bx and By given explicitly by Equas. 28 and 29, and n=1.49) into
Equas. 9 through 12 in order to find the emerging A-rays. Then,
Equas. 13 through 16 are used to find .delta.z'(x,y)/.delta.x and
.delta.z'(x,y)/.delta.y. Next, like Equa. 17, the following third
order polynomial expression is used to represent z'(x,y).
where
g.sub.1.ident.x g.sub.2.ident.y g.sub.3.ident.x.sup.2
g.sub.4.ident.x.multidot.y g.sub.5.ident.y.sup.5
g.sub.6.ident.x.sup.3 g.sub.7.ident.x.sup.2.multidot.y
g.sub.8.ident.x.multidot.y.sup.2 g.sub.9.ident.y.sup.3 (31)
Finally, M, b and a are computed from Equas. 22, 23 and 26. The
results of these calculations appear below. Note that the elements
of the a-vector are the coefficients of the polynomial expression,
Equa. 30, for z'(x,y). It is to be noted that, although the
polynomial expression for z'(x,y) used in this example is only of
order 3 for ease of illustration, the equations and methods of
solution are easily extended to higher order polynomials. Programs
to solve the resulting equations can be written using mathematical
software available for use with personal computers. Of course, as
the order of the polynomial increases, so too do the sizes of the
M-matrix, and the b and a-vectors. For instance, associated with a
polynominal of order 5 is a M-matrix having 20.times.20 elements,
and b and a-vectors each having 20 elements. ##EQU15##
The expression for z'(x,y) defined by Equa. 30 is determined only
up to an arbitrary constant. When one considers that a machining
operation generally is employed to reshape surface z(x,y) to the
new modified surface z'(x,y) and that the machine has to be
programmed to remove "positive" amounts of material, one realizes
that the arbitrary constant has to be large enough to shift surface
z'(x,y) so that, when shifted, its value can never be greater than
z(x,y). For the example, z'(x,y) values are shifted in the negative
z-direction by 0.011247 mm to satisfy this condition. The depth of
cut to modify z(x,y) to the correcting surface is labelled CUT, and
is expressed Equa. 34. Numerical values of 1000*CUT (in microns) at
1 mm spacings in the x & y directions are given in Equa.
35.
4.2.5 Check that the New Surface Corrects Aberrations
In order to realize how well the new surface z'(x,y), described by
the polynomial expression (Equa. 30) with the a-coefficients (Equa.
33), corrects the original optical aberrations, the following
calculations are done: 1) find .delta.z'/.delta.x and
.delta.z/.delta.y from Equa. 30 at all the measurement sites, and
2) use the values just found to find the x, y and z coordinates of
N" from Equa. 1 & 2 where N" represents the normal to z'(x,y)
[see Equa. 30]. Next rewrite Equa. 11 in the following form which
takes into account the labelling for the new correcting
surface:
B" represents the emerging rays after the surface has been
corrected and is the parameter currently being sought. A, which is
invariant with changes to the optical surface, is already known
having been obtained previously by the methods described in the
paragraph preceeding Equa. 30. Likewise N" is known, having been
found by the methods outlined in the paragraph just above. At this
stage, .lambda." is not yet determined; however, it is determined
by the following methods.
First, from Equa. 9 find:
Since the vector product N".times.N"=0, from Equa. 36 find that
N".times.B"=n*(N".times.A). Therefore, Equa. 37 can be
rewritten.
Since n, N" and A are now known parameters, Equa. 38 gives the
solution for .beta.".
Now that solutions for .beta." are found, the corresponding
solutions for .alpha." and .lambda." are found from modified forms
of Equas. 10 & 12 which appear as . . .
B" can now be found from Equa. 36 by substituting the now known
values for n, A, .lambda." and N". In Equas. 41 & 42, the x
& y components of B", which are written as B"x and B"y, appear
multiplied by 1000 which makes the resulting directional units in
milliradians. ##EQU17##
A measure of the degree of collimation of all the B"-rays is its
root-mean-square value, labeled B"rms. It is found by taking the
square root of the average of all 49 values of B"x.sup.2
+B"y.sup.2. The value of B"rms=0.00033 radians. That it is
considerably less than the rms value for the original rays, which
was Brms=0.00148 radians, demonstrates the success of the
algorithms and methods.
With polynomial expansions for z'(x,y) having orders higher than 3
as in this current example, the corresponding size of B"rms is even
less than the value given above. Furthermore, considering that
20/20 visual acuity implies resolving lines spaced 0.00029 radian
apart (i.e. 1 minute of arc), the new correcting surface in this
example is shown to improve visual acuity to very nearly 20/20.
The centroids of the focal spots of the wavefront sensor pattern
for the case of the corrected B"-rays from the new correcting
surface z'(x,y) are shown in FIG. 5. When compared to FIG. 4, which
shows the pattern for the original aberrating optical surface, FIG.
5 shows rays that are much better collimated.
4.3 Fabrication of Aberration Correcting Surfaces on Lenses and
Corneal Tissue
4.3.1 Diamond Point Machining
Since the surface contours of aberration-correcting contact lenses
described by z'(x,y) are more complex than the spherical or
toroidal surfaces of conventional contact lenses, the position of a
cutting tool needed to generate the z'(x,y) surfaces has to be
controlled in a unique way. A programmable computer controlled
single point diamond turning (SPDT) lathe, shown in FIG. 6, has
been used to generate the surfaces of aberration-correcting contact
lenses. The lathe 30 has two moving subassemblies: 1) a low
vibration air bearing spindle 31, and 2) x-z positioning slides 32.
Contact lens 33, which is held and centered on the end of spindle
31, is rotated by the spindle while a fine diamond tool 34 is moved
by the positioning slides 32 with sub-micron resolution both
perpendicular to (i.e. tool transverse scan in x-direction), and
parallel to (i.e. tool cutting depth in z-direction) the direction
of the spindle's axis. Smooth tool motion is achieved by the
preferred means of using hydrostatic oil-bearing or air-bearing
slideways. A preferred means for precise slide positioning is
achieved by using computer controlled piezoelectric drivers or
precise lead screw drivers (not shown). Since the machine must be
completely free of both internal and external vibrations, both
lathe 30 and x-z slides 32 are secured to a pneumatically isolated
table top 35 which rests on granite base 36.
During lens machining, a computer 37 receives synchronous signals
from spindle 31 and controls the movement of the x-z translation
slides 32 along a programmed trajectory that is synchronized with
the rotational position of the spindle. The motion of the
x-translation slide (i.e. perpendicular to the spindle axis)
generally is at a uniform speed as with an ordinary lathe. However,
the requirement of forming a non-axially symmetric lens surface
(i.e. z'(x,y) shape), when using a high speed lathe, requires a
unique control method for positioning the z-translation slide (i.e.
controls cutter movement parallel to the spindle axis and,
consequently, the depth of cut on the lens surface which rotates
rapidly with respect to the cutter). The z-translation slide must
be rapidly and precisely located in accord with both the
x-translation slide location, and the rotational position of the
spindle. The preferred means of rapid and precise positioning of
the z-translation slide is by the utilization of computer
controlled piezoelectric drivers.
Finally, mounting the optical element firmly on a supporting block
before placement on the end of the spindle is extremely important
to avoid distorting the optical surface during machining. Care also
must be taken to avoid distorting the lens blank when securing it
to the supporting block; otherwise, the surface will warp after
removing it from the block following final surfacing.
As an alternative to a precision lathe, custom contact lenses can
also be machined by an x-y-z contour cutting machine. With a
machine of this type, the lens is held stationary and figured by a
cutting tool driven in a precise x-y raster scan while the depth of
cut in the z-direction is controlled by a computer often with
positional feedback provided by an interferometer. Such an x-y-z
contour cutting machine is described by Plummer et al. at the 8th
International Precision Engineering Seminar (1995), and published
in those proceedings by the Journal of American Society of
Precision Engineering, pp. 24-29.
4.3.2 Surfacing by Ablation using an Ultraviolet Laser
Laser ablation of corneal tissue currently is used to reshape the
anterior corneal surface in order to correct near and
farsightedness and astigmatism in the human eye. Laser tissue
ablation also can be used to make more subtle corrections to the
corneal surface than are now performed. Such corrections are needed
to correct the eye's higher order optical aberrations. Therefore,
laser surface ablation is a preferred method of contouring corneal
tissue in order to correct the eye's higher order optical
aberrations.
Also, as an alternative to machining optical plastic contact lenses
with a fine diamond point tool, laser machining, similar to the
laser ablation technique used to reshape the surface of the cornea,
is also possible. It is likely that the most useful lasers for this
purpose will prove to be those emitting ultra-violet light which
ablate material away from surfaces, not by thermal heating or
melting, but by rapid disruption of chemical bonds. Excimer lasers
and solid state frequency-tripled (or frequency-quadrupled) Nd-YAG
lasers are now proving to be useful both for machining plastics and
ablating human tissue. These same lasers may be considered for
precise surfacing operations needed for the fabrication of
aberration correcting contact lenses or advanced refractive
surgery.
4.4 Alternate Methods for Fabrication of Aberration Correcting
Lenses
In addition to diamond point machining and laser ablation methods,
there are other conceivable ways of fabricating an optical lens
that can compensate for the eye's irregular optical aberrations.
Any successful fabrication method must be capable of either
precisely forming the lens surface to the required z'(x,y) shape,
or meticulously varying the refractive index over the surface in
order to bend the light rays to correct the aberrations. Described
below are several alternate lens fabrication methods.
4.4.1 Mechanical Force Thermal Molding
If the lens is a thermoplastic, its surface may be formed to match
that of a heated die. This method follows from the techniques used
to form plastic parts by injection or compression molding. Since
the desired topography of the lens surface generally is complex,
the surface of the heated die is required to have a corresponding
complex shape. When normally forming a plastic lens by the methods
of injection or compression molding, the die which determines the
shape of the lens is made of a single piece of metal and has a
permanent surface shape. To form the customized and complex lens
surfaces needed to correct the irregular aberrations of the human
eye, it is necessary to construct and utilize dies with variable or
"adaptive" surfaces. In the field known as "adaptive optics", such
variable surfaces (anecdotally referred to as "rubber mirrors")
have been formed and controlled using computer-derived input
signals that drive electromechanical fingers which press against a
deformable metal surface.
Such an arrangement, which is called here a variable segmented die,
is shown in FIG. 7. The housing for the die 40 provides a number of
channeled holes to hold a corresponding number of mechanical
fingers 41 which are moved upwards or downwards in the die housing
40 by electronic drivers or actuators 42. Mechanical fingers 41
press against a continuous deformable metal surface 43 which
contacts the thermoplastic lens surface and establishes its final
shape. As an alternative to using a customized heated die with its
surface formed by electromechanical fingers pressing in a
controlled way against a deformable metal surface, an arrangement
of close-packed, electronically actuated mechanical fingers without
a deformable metal surface may be used. With this alternative
arrangement, each mechanical finger in the array would press
directly on the surface of the thermoplastic lens blank in order to
form its surface as required.
4.4.2 Photolithography and Etching
A layer of photoresist is spun onto the glass or plastic lens
substrate, and selectively exposed using either an optical scanner
or an electron beam scanner. The exposure extent over regions of
the surface of the photoresist is properly matched with the desired
surface contour of the finished lens. The photoresist is developed
chemically, thereby being selectively prepared for subsequent
etching over those areas having previously received the most
illumination. The surface of the glass or plastic lens substrate is
then etched using reactive ion etching or chemically assisted
ion-beam etching where the depth of the etch is determined by the
extent of illumination during the previous exposure of the
photoresist. It is noted that forms of PMMA (i.e. polymethyl
methacrylate), which is a widely used optical plastic used for
making contact lenses, are used as photoresists. Therefore, one can
imagine tailoring the surface of a PMMA lens, in order to correct
an eye's higher-order aberrations, directly without needing to use
a separate photoresist layer over the lens substrate.
4.4.3 Thin Film Deposition:
There are various techniques now employed to add thin layers of
various materials to the surfaces of glass and plastic optical
lenses. The current purposes for applying such thin films to lenses
are: 1) to limit the transmission of light, 2) to reduce surface
reflections, and 3) to protect surfaces from scratching and
abrasion. Some of the methods employed include dip coating, spin
coating, evaporative coating, spraying, and ion sputtering. To
modify these existing methods for the purpose of making a lens
useful to correct the higher-order aberrations of the human eye
will require refinements that allow the deposition of layers with
thicknesses that vary selectively over the surface of the lens
substrate in order to achieve the required z'(x,y) shape.
In addition to the methods now used to deposit thin films on
optical surfaces, one can imagine other methods that may prove
possibly more useful. Particularly useful could be a method of
injecting from a fine nozzle or series of fine nozzles, which are
in proximity to a surface, controlled amounts of transparent liquid
materials that permanently bond when deposited on the surface.
Scanning these controllable material-depositing nozzles over the
surface of an optical substrate can result in building up on the
optical substrate a custom surface contour that meets the
requirements of correcting the higher-order optical aberrations of
a subject's eye. Similar to what is conceived above is the
operation of ink-jet devices used in computer printers.
4.4.4 Surface Chemistry Alteration:
Surfaces of flat glass disks have been implanted with certain ionic
impurities that result in index of refraction changes that can vary
radially from the center to the edge of the disk, or axially over
the depth of the disk. Such methods are utilized in the fabrication
of so-called "gradient index" optical elements, and flat disks are
available commercially that behave as ordinary positive or negative
lenses. Although plastic gradient index lenses have not yet been
made, one can imagine altering the refractive index of a plastic
subsurface by certain techniques. For example, the exposure of a
plastic surface to ultraviolet light can alter the polymerization
of subsurface macromolecules and, thereby, change the index of
refraction in the subsurface region. For another example, the
methods now used to imbibe dye molecules from solution into plastic
surfaces in order to tint ophthalmic lenses may also be effective
in changing the index of refraction in subsurface layers. In the
future, it is conceivable that the methods of surface chemistry
alteration, which are used now to fabricate "gradient index"
optical elements, can be refined for both glass and plastic
materials in order to make them useful in the fabrication of
adaptive optical lenses for correcting the higher-order aberrations
of the human eye.
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U.S. Patent Documents: U.S. Pat. No. 5,777,719 issued Jul. 7, 1998
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