Note: Descriptions are shown in the official language in which they were submitted.
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SYSTEMS AND METHODS OF PHASE DIVERSITY WAVEFRONT SENSING
BACKGROUND AND SUMMARY
[0001] Field.
[0002] This invention pertains to the field of wavefront measurements, and
more
particularly to systems and methods of measuring a wavefront of light using a
phase diversity
wavefront sensor.
[0003] Description.
[0004] A number of systems and methods have been developed for measuring a
wavefront
of light. Such wavefront measurements have been employed in a number of
applications,
including ophthalmic applications such as measuring aberrations of an eye, and
measuring
surfaces of objects such as contact lenses.
[0005] One wavefront sensor that has been employed in a number of systems for
various
wavefront sensing applications is the Shack Hartmann wavefront sensor (SHWS).
A SHWS
includes an array of lenslets which image focal spots onto a detector array.
SHWS's have
been employed in a variety of ophthalmic and metrological applications.
[0006] However, a SHWS has some limitations in certain applications.
[0007] For example, with a SHWS, the wavefront is expected to produce a single
local tilt.
In general, an SHWS has difficulty measuring wavefronts with discontinuities.
However, in
some applications, and particularly in some ophthalmic applications, the
wavefront may have
multiple tilts, which may produce multiple focal spots. For example, such
discontinuities can
be produced by multi-focal optical devices, including multifocal contact
lenses and
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multifocal intraocular lenses 0014 W. Neil Charman et al., "Can we measure
wave
aberration in patients with diffractive IOLs?," 33 JOURNAL OF CATARACT &
REFRACTIVE
SURGERY No. 11, p. 1997 (Nov. 2007) discusses some problems in using a SHWS to
make
wavefront measurements of a patient with a diffractive IOL. Charman notes that
when a
measurement is taken on an eye that has been implanted with a diffractive IOL,
the lenslets of
the SHWS will produce multiple images and the detector will record multiple
overlapping
spot patterns. So, it is difficult at best for a SHWS to measure wavefronts
produced by
multifocal optical elements, such as diffractive IOLs.
[0008] Another limitation of the SHWS pertains to its limited dynamic range.
For example,
to measure ophthalmic aberrations of a human eye over the wide range presented
by the
human population, as a practical matter one needs to employ an adjustable
optical system in
conjunction with the SHWS so that operation of the SHWS can be maintained
within its
dynamic range. This can add to the complexity and cost of the measurement
system, and
requires alignment that can reduce the measurement precision of the
instrument.
[0009] Another type of wavefront sensor is a phase diversity wavefront sensor
(PDWS),
also sometimes referred to as a curvature sensor. A PDWS may be used to
analyze
wavefronts at two or more planes that are generally orthogonal to the
direction of propagation
of an optical beam. In general, a PDWS measurement system makes measurements
via an
optical system that is capable of imaging two or more planes at once, to
minimize or
eliminate the effects of any time-varying changes in the optical beam. Graves
et al. U.S.
Patent 6,439,720 describes a measurement system that includes a PDWS. Early
PDWS
systems employed a relatively complex arrangement of beam splitters and/or
optical delays to
generate the necessary images.
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[00010] In 1999, Blanchard, P.B and Greenaway, A.H., "Simultaneous Multi-plane
Imaging
with a Distorted Diffraction Grating," APPLIED OPTICS (1999) ("Blanchard")
disclosed the
use of a diffractive optical element (DOE) in a PDWS. As disclosed by
Blanchard, the DOE
uses local displacement of lines in a diffraction grating to introduce
arbitrary phase shifts into
wavefronts diffracted by the grating into the non-zero orders to create
multiple images of the
incident light. In Blanchard's arrangement, a diffraction grating having a
quadratic
displacement function is employed in conjunction with a collocated single lens
to alter the
optical transfer function associated with each diffraction order such that
each order has a
different degree of defocus. Greenaway et al. U.S. Patent 6,975,457 and
Greenaway et al.
U.S. Patent Application Publication 2006/0173328 describe further details of a
PDWS that
includes a DOE.
[00011] Otten III et al. U.S. patent 7,232,999 discloses the use of a PDWS
with a DOE for
determining the characteristics of an infrared wavefront produced by a laser.
Slimane Djidel,
"High Speed, 3-Dimensional, Telecentric Imaging," 14 OPTICS EXPRESS No. 18 (4
Sep. 2006)
describes design, testing and operation of a system for telecentric imaging of
dynamic objects
with a single lens system. However, the procedure described therein is not
extensible to
more complicated configurations.
[00012] Nevertheless, these references are not generally directed to
applications where there
is speckle and/or discontinuities or large aberrations in the wavefront, such
as may be the
case in many ophthalmic applications, including the measurement of IOLs,
multifocal
contact lenses, etc., and eyes or optical systems that include such devices.
Furthermore, these
references do not provide a generalized design method for incorporating a PDWS
into more
complicated optical systems.
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[00013] It would be desirable to provide an ophthalmic measurement instrument
that utilizes
the benefits of a PDWS, alone or in conjunction with a SHWS. It would further
be desirable
to provide such an instrument that can measure wavefronts with speckle and/or
discontinuities or large aberrations in the wavefront. More particularly, it
would be desirable
to provide such an instrument that can perform wavefront measurements for
systems that
include a multifocal element, such as an intraocular or contact lens that is
either a refractive
multifocal lens, a diffractive multifocal lens, or a diffractive monofocal
lens. It would also be
desirable to provide a generalized method of designing a measurement system
including a
PDWS.
[00014] In one aspect of the invention, a phase diversity wavefront sensor
comprises: an
optical system including at least one optical element for receiving a light
beam; a diffractive
optical element having a diffractive pattern defining a filter function, the
diffractive optical
element being arranged to produce, in conjunction with the optical system,
images from the
light beam associated with at least two diffraction orders; and a detector for
detecting the
images and outputting image data corresponding to the detected images, wherein
the optical
system, diffractive optical element, and detector are arranged to provide
telecentric, pupil
plane images of the light beam.
[00015] In another aspect of the invention, a method is provided for measuring
a wavefront
of an optical system including a multifocal element. The method comprises:
providing a light
beam to a lens, the lens being a refractive multifocal lens, a diffractive
multifocal lens, or a
diffractive monofocal lens; directing light from the lens to a phase diversity
wavefront
sensor, comprising an optical system including at least one optical element
for receiving a
light beam, and a diffractive optical element the shape of which is defined by
a filter
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function, the diffractive optical element being arranged to produce in
conjunction with the
optical system images of the light beam associated with at least two
diffraction orders; and a
detector for detecting the images and outputting image data corresponding to
the detected
images; and measuring the wavefront of the light from the lens using the image
data output
by the detector.
[00016] In yet another aspect of the invention, a method is provided for
measuring a
wavefront of an object having first and second surfaces. The method comprises:
providing a
light beam to the object; directing light from the lens to a phase diversity
wavefront sensor,
the lens being a refractive multifocal lens, a diffractive multifocal lens, or
a diffractive
monofocal lens, the phase diversity wavefront sensor comprising an optical
system including
at least one optical element for receiving a light beam, and a diffractive
optical element the
shape of which is defined by a filter function, the diffractive optical
element being arranged
to produce in conjunction with the optical system images of the light beam
associated with at
least two diffraction orders; and a detector for detecting the images and
outputting image data
corresponding to the detected images; and simultaneously measuring the first
and second
surfaces of the object using the image data output by the detector.
[00017] In still another aspect of the invention, a method is provided for
designing a phase
diversity wavefront sensor. The method comprises: providing one or more
analytic solutions
for paraxial equations that govern an optical configuration of the phase
diversity wavefront
sensor; providing a set of input design parameters for the phase diversity
wavefront sensor;
generating a set of output values from the analytical solutions and the input
design
parameters; and determining whether the output parameters meet a viability
threshold.
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[00018] In a further aspect of the invention, a phase diversity wavefront
sensor comprises:
an illuminating optical system for delivering light onto a retina of an eye; a
receiving optical
system for receiving light reflected by the retina, the receiving optical
system comprising a
diffractive optical element including a diffraction pattern defining a filter
function, the
diffractive optical element being arranged to produce, in conjunction with the
optical system,
at least two images from the light beam associated with at least two
diffraction orders; a
detector for detecting the at least two images; a memory containing
instructions for executing
a Gerchberg-Saxton phase retrieval algorithm on data produced by the detector
in response to
the detected images; and a processor configured to execute the Gerchberg-
Saxton phase
retrieval algorithm so as to characterize a wavefront produced by the
reflected light.
BRIEF DESCRIPTION OF THE DRAWINGS
[00019] FIG. 1 illustrates the use of a diffractive optical element (DOE) in a
phase diversity
wavefront sensor (PDWS).
[00020] FIG. 2 illustrates an intensity image produced by the PDWS of FIG. 1.
[00021] FIG. 3 illustrates another configuration of a PDWS.
[00022] FIG. 4 illustrates one embodiment of diffraction grating.
[00023] FIG. 5 illustrates an intensity image produced by the PDWS of FIG. 3.
[00024] FIG. 6 illustrates operation of one embodiment of a Gerchberg-Saxton
(GS)
algorithm.
[00025] FIG. 7 illustrates propagation from one measurement plane to the next.
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[00026] FIG. 8 illustrates the numerically calculated defocus versus iteration
number in a
GS algorithm for different pupil diameters.
[00027] FIG. 9 plots the number of iterations in a GS algorithm required to
reduce the
defocus error to less than 0.01 diopters versus pupil diameter.
[00028] FIG. 10 plots the number of iterations in a GS algorithm required for
convergence
versus sample plane separation for a given beam diameter.
[00029] FIG. 11 illustrates the numerically calculated defocus versus
iteration number in a
GS algorithm for different pupil diameters in the case of an irradiance
pattern where speckle
is introduced.
[00030] FIG. 12 plots the number of iterations required to reduce the defocus
error to less
than 0.01 diopters versus pupil diameter for a speckled beam, compared to a
beam without
speckle.
[00031] FIGs. 13A-C illustrate basic ophthalmic aberrometer designs for SHWS
and PDWS
sensors.
[00032] FIG. 14 illustrates a simplified design of a PDWS with a large dynamic
range.
[00033] FIG. 15 illustrates a process of designing a measurement system that
includes a
PDWS.
[00034] FIG. 16 illustrates how the process of FIG. 15 establishes design
tradeoffs by
comparing design points.
[00035] FIG. 17 illustrates how a PDWS can be used to measure both surfaces of
a contact
lens.
[00036] FIGs. 18A-C illustrate the use of a PDWS in an ophthalmic measurement
application.
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[00037] FIG. 19 illustrates a block diagram of one embodiment of an ophthalmic
aberrometer that includes a PDWS.
DETAILED DESCRIPTION
[00038] FIG. 1 illustrates the use of a diffractive optical element (DOE) in a
phase diversity
wavefront sensor (PDWS) 100. PDWS 100 includes an optical element 110, a
detector 120,
and a processor 130. Optical element 110 includes a diffractive optical
element (DOE) (e.g.,
a diffraction grating) 112 collocated with optical element 114 with positive
focal power.
Although shown in transmission mode, optical element 110 may alternately be
used in
reflection where diffraction grating 114 is collocated with optical element
114 comprising a
mirror.
[00039] In the illustrated embodiment, optical element 114 is a lens, and
diffractive grating
112 is disposed on a surface of lens 114. Alternatively, diffractive grating
112 may be
incorporated inside lens 114 or be formed from the material used to form lens
114. In some
embodiments, lens 114 and diffractive grating 112 form a single DOE, where
lens 114 is
itself a DOE, for example, disposed on a same surface or an opposite surface
as diffractive
grating 112. In yet other embodiments, lens 114 and grating 112 are separate
elements that
touch one another or are separated by a relatively small distance. Element 114
could be
refractive, diffractive or reflective.
[00040] Detector 120 may be a charge coupled device (CCD).
[00041] In one embodiment, diffraction grating 112 is distorted by a quadratic
filter function
so that optical element 110 introduces an optical power that depends upon the
diffraction
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order. Optical element 110 produces angularly displaced beams with different
focal power.
The combination of diffraction grating 112 and lens 114 yields a net optical
power given by:
1 1 R2
(1) P = __ = + ______ = P + P
LENS GRATING
fTOTAL f 2m W20
where m is the diffraction order of diffraction grating 112, R = the aperture
radius of
diffraction grating 112 and W20 is a standard defocus term specifying the
phase shift from
center to edge of the optic. This is related to the quadratic distortion in
the grating as
specified by Blanchard. Note that the grating period in such distorted
gratings is not
constant, but can still be specified in terms of an average period at the DOE
center. This
grating period is the average distance between the lines in the grating and,
together with the
wavelength of the incident light, determines the diffraction angle of the
diffracted beams, and
hence their separation on the detector array.
[00042] In one embodiment, diffraction grating 112 is distorted by a filter
function that is
non-quadratic and has non-mixed symmetry.
[00043] In the case of illumination of DOE 110 by plane wave 10, it is clear
that each order
produces a focus on either side of the detector plane.
[00044] In PDWS 100, detector 120 is located at the focal plane for the 0th
order beam and is
referred to as an "image plane PDWS." In that case, FIG. 2 illustrates an
intensity image
produced by PDWS 100. This arrangement produces a real image at the +1
diffraction order,
a virtual image at the -1 diffraction order, and a far field pattern at the 0
diffraction pattern.
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As can be seen in FIG. 2, this produces a bright spot 0th order beam, and
dimmer spots for the
+1 and -1 diffraction orders.
[00045] Data acquisition may be accomplished by two-dimensional digitization
of the
intensity image at detector 120. The image data is then supplied to processor
130 for further
analysis to measure the wavefront of plane wave 10.
[00046] FIG. 3 illustrates another configuration of a PDWS 300. PDWS 300
comprises an
optical element (e.g., a lens) 310, a diffractive optical element (e.g., a
diffraction grating)
320, a camera or detector 330, and a processor 340. Detector 330 may comprise
a charge
coupled device (CCD). PDWS 300 possesses certain characteristics that may be
beneficial
for measuring wavefronts in ophthalmic applications, as will be discussed in
greater detail
below. Associated with processor 340 is memory 345 containing instructions for
executing a
phase retrieval algorithm on data produced by detector 330.
[00047] FIG. 4 illustrates one embodiment of diffraction grating 320. In one
embodiment of
FIG. 4, diffraction grating 320 comprises a distorted grating that is
distorted by a quadratic
filter function. This is also known as an off-axis Fresnel lens.
[00048] In one embodiment, diffraction grating 320 is distorted by a filter
function that is
non-quadratic and has non-mixed symmetry.
[00049] FIG. 5 illustrates an intensity image produced by PDWS 300. In
contrast to PDWS
100, which is an example of an Image Plane PDWS, PDWS 300 forms real images of
the
beam at both sample planes and at the measurement plane (the Pupil Plane).
Accordingly,
PDWS 300 is referred to as a "Pupil Plane PDWS."
[00050] As shown in FIG. 3, PDWS 300 forms images of the beam 350 at different
sample
locations, and these images are laterally displaced at camera 330 so that they
can be
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simultaneously acquired. PDWS 300 can be thought of as producing multiple
object planes
(also referred to as "observation planes" or "sample planes') that are imaged
onto camera
330. In particular, object plane u_1 is imaged onto the -1th order beam,
object plane u0 is
imaged onto the 0th order beam, and object plane u+i is imaged onto the +1th
order beam at
camera.
[00051] Although FIG. 3 illustrates an example with a converging beam, a
collimated beam
or a diverging beam may be employed in a particular application. Also,
although FIG. 3
illustrates three "observation planes" it should be understood that more
observation planes
corresponding to additional diffraction orders can be employed and that only
two observation
planes are necessary in many applications. On the other hand, having a
multitude of
observation planes can provide a greater dynamic range, greater sensitivity,
improved ability
to discern waves with multiple wavefronts.
[00052] Data acquisition may be accomplished by two-dimensional digitization
of the
intensity values detected by camera 330. The detected intensity data may then
be analyzed
by processor 340 to determine the phase distribution that produces the
intensity measured in
all planes, as will now be explained in detail.
[00053] Knowledge of the sampled intensity profiles, the locations of the
sample planes, and
the wavelength of the beam are generally sufficient to determine the
phasefront of the beam.
One phase retrieval method that has been applied to PDWS data is to derive
solutions to the
Intensity Transport Equation (ITE). Phase retrieval via the ITE is fast and
analytic.
[00054] Unfortunately the application of ITE analysis to highly aberrated
beams may be
problematic. M. R. Teague, 73 JOSA No. 11, pg. 1434 (1983) derived the ITE
from the wave
equation expressly for the phase retrieval problem. He showed that for a beam
of intensity, I,
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wavefront, 4), and wave number, k=27c/1, then its transverse derivatives and
its axial derivative
are related by:
(2) k ¨0I¨ IV 2c0 ¨ V/ = V c0 k I-1¨ I1
Oz z _1¨ z1
where
(3) U(x, y) = VI(x, y) elP(X,Y)
[00055] Since the axial derivative is not known, it is approximated by the
finite difference
between the intensity measurements along the propagation direction as shown in
EQN. 2
above. This approximation fails for beams with aberrations large enough to
significantly
change the beam size between the sample planes. As such properties may be
found in beams
in ophthalmic applications, the use of ITE-based phase retrieval methods is of
limited utility,
for example, for a PDWS employed in an ophthalmic aberrometer.
[00056] Accordingly, to increase the phase retrieval accuracy for beams with
large
aberrations, and thereby to provide a solution for employing PDWS 300 in
ophthalmic
applications ¨ such as in an ophthalmic aberrometer ¨ in PDWS 300 processor
340 performs
a Gerchberg-Saxton (GB) phase retrieval algorithm using the intensity data
from camera 330.
The GS method does not require knowledge of the axial intensity derivative,
but uses all
intensity measurements to numerically calculate the phase front. The GS method
is an
iterative process where known intensity measurements are used with wave
propagators to
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estimate the intensity and phase at the next measurement plane. Before each
successive
propagation step, the predicted intensity is replaced with the measured
intensity.
[00057] FIG. 6 illustrates operation of one embodiment of the GS algorithm. In
a first step,
processor 340 estimates, or guesses, (I)(x,y). In a next step, processor 340
takes the latest
estimate ofq)(x,y) and propagates it to the next measurement plane. Then,
processor 340
replaces the amplitude of the propagated field with the square root of the
intensity
measurement at that plane. Processor 340 then propagates this data to the next
measurement
plane, and the process is repeated for all measurement planes until the
propagated intensity
matches the measured sufficiently well (e.g., the difference is less than a
defined threshold).
If necessary, the process may proceed from the [Li measurement plane, to the
IA measurement
plane, to the IAA measurement plane, and back to the [to measurement plane,
then to the [Li
measurement plane, etc., until convergence is reached.
[00058] In one embodiment processor 340 employs a Rayleigh-Sommerfeld
propagation
integral to propagate from one measurement plane to the next. FIG. 7
illustrates this
propagation. Given the data Ma,13) at a first measurement plane ji, then the
data is
propagated to a second measurement plane, [Li, to produce propagated data
i.t2(x,y) as follows:
(4) u2(x,y)= ffui(a,fl) z12 eikr12 dadfl
l21\ 12
[00059] The inventors have investigated the efficacy of the iterative GS phase
retrieval
method in ophthalmic instruments where large dynamic range in defocus and the
presence of
speckle make phase retrieval with standard methods based on the intensity
transport equation
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difficult. Simulated PDWS data covering a typical range of ophthalmic defocus
aberrations
with a standard PDWS configuration were generated using the Rayleigh-
Sommerfeld
propagation integral equation. The data was processed using the GS method and
the
following parameters were varied to study the robustness of the method and its
rate of
convergence: input pupil diameter; sample plane spacing; and irradiance
characteristics.
Only beams with spherical wavefronts and intensity distributions at z = 0 that
are zero
outside a circular pupil were studied. Three intensity measurements were used.
A
wavelength of 635 nm was assumed. Speckled beams were simulated by imposing a
random
phase distribution with amplitude several radians on a uniform beam and
propagating several
millimeters. The size of the speckle cells of the resulting intensity
distributions averaged
about 1 mm.
[00060] FIG. 8 illustrates the numerically calculated defocus versus iteration
number for
different pupil diameters. FIG. 9 plots the number of iterations required to
reduce the
defocus error to less than 0.01 diopters versus pupil diameter. FIGs. 8 and 9
show that
convergence is rapid for small diameter beams but is much slower as the beam
diameter
increases. The number of iterations required to achieve a specified level of
defocus accuracy
increases approximately exponentially with input pupil diameter for fixed
sample spacing.
[00061] FIG. 10 plots the number of iterations required for convergence versus
sample plane
separation for a given beam diameter. It can be seen in FIG. 10 that the
convergence rate
improves with sample plane separation. For a given beam size, the number of
iterations
required to achieve a specified level of defocus accuracy decreases as the
reciprocal of the
sample plane spacing.
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[00062] FIG. 11 illustrates the numerically calculated defocus versus
iteration number for
different pupil diameters in the case of an irradiance pattern where speckle
is introduced.
FIG. 12 plots the number of iterations required to reduce the defocus error to
less than 0.01
diopters versus pupil diameter for a speckled beam (lower plot), compared to a
beam without
speckle (upper plot). In the case of a speckled beam, the number of iterations
required to
achieve a specified level of defocus accuracy increases approximately
quadratically with
input pupil diameter for fixed sample spacing, rather than exponentially, as
is the case with
beams that do not include speckle. Beneficially, for beams having relatively
large beam
diameters (e.g., beam diameters greater than or equal to 1.5 millimeters or
greater than or
equal to 2 millimeters), this significantly reduces the number of iterations
required for beams
containing large numbers of speckle cells, as is typical in ophthalmic
aberrometers.
[00063] In PDWS 300, the dynamic range and sensitivity can be controlled by
proper
selection of the sample plane spacing and the number of bits of digitization
of the CCD in
camera 330. Meanwhile, the resolution is controlled by the magnification and
the pitch of the
pixels in camera 330. Beneficially, PDWS 300 provides a wide dynamic range so
as to
accommodate a wide range of aberrations in the input wavefront without the
need to move or
adjust any optical elements, thus simplifying the construction of an
ophthalmic measurement
instrument. Beneficially, in one embodiment PDWS 300 is capable of measuring
the
wavefronts of beams with at least + 3 diopters of defocus. Further
beneficially, in one
embodiment PDWS 300 is capable of measuring the wavefronts of beams with at
least + 5
diopters of defocus. Even further beneficially, in one embodiment PDWS 300 is
capable of
measuring the wavefronts of beams with at least + 10 diopters of defocus.
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[00064] PDWS 300 includes a number of features that are desirable for an
ophthalmic
measurement system. Pupil plane imaging provides a real image of the pupil and
accommodates variability in the location, size and shape of a human pupil when
making
aberrometer measurements, especially because the location of a patient's eye
is generally not
well controlled. Pupil Plane Imaging is also beneficial in resolving the phase
of a speckled
beam, or a wavefront having one or more discontinuities.
[00065] Also beneficially, PDWS 300 may employ telecentric imaging.
Telecentric imaging
provides equally spaced sample planes, and provides equal magnification for
all images.
Telecentric imaging simplifies the alignment, calibration, and data processing
of PDWS 300.
Further details of the telecentric arrangement will be provided below.
[00066] FIGs. 13A-C illustrate basic ophthalmic aberrometer designs for SHWS
and PDWS
sensors. Although not shown in FIGs. 13A-C, in practice the ophthalmic
aberrometer will
include a light projection system for creating the light beam and directing it
to an eye or other
object that is being measured. For example, the wavefront u0 may be an image
of a
wavefront at a pupil or corneal surface of an eye under examination.
[00067] FIG. 13A illustrates an exemplary design for an ophthalmic aberrometer
1300A
employing a SHWS 1310A. SHWS 1310A includes a lenslet array 1312A and a
camera, or
pixel array 1314A, also called a detector array. The design employs a Badal
Relay Imager
1320A including two lenses 1322A and 1324A. A processor 1350A processes data
produced
by camera 1314A.
[00068] With the SHWS 1310A, both the spatial resolution and the dynamic range
are
correlated to the dimension of the lenslets in lenslet array 1312A. The
optical system
typically demagnifies the pupil image to fit on SHWS 1310A and the distance
between lenses
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1322A and 1324A is adjusted to add defocus to compensate the incoming
wavefront so that it
lies within the dynamic range of SHWS 1310A. Preservation of the optical phase
front is
important with SHWS 1310A, and image quality is generally a secondary
consideration in
the optical design. The sensitivity of SHWS 1310A is set by the lenslet focal
length and the
pixel size in camera 1314A and is adjusted to give a predetermined
sensitivity. The
sensitivity and spatial resolution requirements typically limit the dynamic
range of SHWS
1301A to a few diopters. However, in aberrometer 1300A, the system can be
dynamically
adjusted to produce a larger effective dynamic range by moving one or both of
the lenses
1322A and 1324A.
[00069] FIG. 13B illustrates an exemplary design for an ophthalmic aberrometer
1300B
where a PDWS 1330B replaces the SHWS 1310A of FIG. 13A. PDWS 1330B includes
lens
1332B, diffraction grating 1333B and camera, or pixel array 1334B, also called
a detector
array. A processor 1350B processes data produced by camera 1334B.
[00070] However, the arrangement of aberrometer 1300B is unnecessarily
complex. Indeed,
an analysis of the paraxial equations shows that telecentric imaging
conditions can be
specified, but the sample plane locations of the + 1 orders and the
constrained location of
diffraction grating 1333B are all nonlinearly related to the spacing between
lenses 1322B and
1324B. Adjustment of Badal Relay Imager 1320B hence requires complex
adjustment of the
position of diffraction grating 1333B. This suggests that such adjustment is
not necessary,
and indeed, significantly it suggests that Badal Relay Imager 1320B can be
omitted.
[00071] FIG. 13C illustrates an exemplary design for an ophthalmic aberrometer
1300C is
tailored for PDWS 1330C. Aberrometer 1330C includes lens 1332C, diffraction
grating
1333C and camera, or pixel array 1334C, also called a detector array. A
processor 1350C
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processes data produced by camera 1334C. By taking advantage of the large
native dynamic
range of PDWS 1330C, and the fact that it is essentially a multiplane imager,
the part count
can be reduced and the moving parts found in FIGs. 13A-B can be eliminated.
[00072] Accordingly and beneficially, ophthalmic aberrometer 1300C provides a
comparable dynamic range to that of 1300A, yet requires no moving elements.
That is, the
positional relationship between all optical elements in ophthalmic aberrometer
1300C
remains constant.
[00073] Turning again to FIG. 3, as noted above, beneficially PDWS 300 employs
telecentric imaging. A generalized design procedure will now be explained for
an
aberrometer including PDWS 300 so as to provide the desired telecentric
imaging.
[00074] FIG. 14 illustrates a simplified design of a PDWS with a large dynamic
range. FIG.
14 shows a first lens 1410, a second lens 1420, a diffraction grating 1430, a
camera 1440, and
a processor 1450. Analytic solutions with Pupil Plane and Telecentric Imaging
and the use of
static optical elements will be explained with respect to FIG. 14.
[00075] In one embodiment, an analytic solution is performed for the paraxial
equations that
govern the particular optical configuration of interest, using ray matrix
analysis, to determine
the proper arrangement to provide telecentric imaging. By first solving the
paraxial
equations analytically, the telecentric solution can be found by imposing the
appropriate
constraints on the general imaging solution; these constraints select the
subset of the general
paraxial imaging solutions with magnification independent of grating order, or
equivalently,
object positions that depend purely linearly on grating order. In one
exemplary but non-
limiting embodiment, the object plane locations for all images depend linearly
on the grating
order and the image magnifications are independent of the grating order for an
optical
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configuration consisting of two lenses followed by a grating as shown in
Figure 13C. The
lens focal lengths are respectivelyfi and f, the grating focal length in first
order is fg, m is the
grating order, s is the distance between the second lens and the grating, t is
the distance
between the lenses and v is the space between the second lens and the detector
array.
Equation 5 shows the general solution for the telecentric pupil plane Lens-
Lens-Grating
PDWS.
S = f
(.-tv f +1)) f 2 nt 4'2
(5) =
mõrr = _
Jfid,4 4,74
[00076] The general telecentric pupil plane imaging PDWS equations shown above
describe
a family of solutions in which s, v and t are related for a given set of lens
and grating focal
lengths. Table 1 below shows representative examples of the family of analytic
paraxial
solutions for the Lens-Lens-Grating configuration of FIG. 14, derived using a
symbolic
manipulator (e.g., MATHEMATICAO) as shown in Equation 5, that provide both
telecentric
and pupil plane imaging for static lens positions for specific values of t and
v. The sample
plane locations um (e.g., [Li , [to , [LA ) are linear in grating order, m,
and the magnification is
independent of grating order, characteristics of a telecentric imaging system.
Note that the
solution with t =fi is a telecentric pupil plane PDWS where the second lens
and grating co-
located; although this looks similar to the image plane sensor, the judicious
positioning of
each optical element provides the additional functionality of the pupil plane
PDWS.
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[00077] TABLE 1
t=f1 t = f t = f = v
s = 0
õ r2
s = ¨ -7- S = I
{' fi
a;31 = Ji1 ¨ ¨
= t2 (s_f 2t.'4 ) m
Mag = 1 ¨ ¨
Mag =
[00078] FIG. 15 illustrates a process 1500 of designing a measurement system
that includes
a PDWS.
[00079] In a first step 1510, the analytical solutions are imported into a
spreadsheet to
explore the performance of the system versus input design parameters.
[00080] Then, in a step 1520, input design parameters are provided. The inputs
may include
the optical configuration, the location of the pupil plane, the desired
dynamic range.
[00081] In a step 1530, outputs are generated based on the analytical
solutions and the input
design parameters. Outputs may include sensitivity, system length, actual
dynamic range, etc.
[00082] In a step 1540, it is determined whether a viable design has been
produced. If not,
then the process returns to step 1520 and new input parameters are provided.
If a viable
deign has been achieved, then a detailed analysis is performed in step 1550.
[00083] FIG. 16 illustrates how the process of FIG. 15 establishes design
tradeoffs by
comparing design points. FIG. 16 plots sensitivity versus pupil plane
location. So, for
example, if the system requires a sensitivity of at least 0.01 diopters and a
stand-off distance
between 73 and 375 mm, as illustrated in FIG. 16, an acceptable performance
range exists
and final detailed ray matrix analysis of this system configuration is
warranted as it is a
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viable design. This design method is beneficial in assisting in the early
rejection of candidate
configurations before significant investment is made in their detailed
analysis; in contrast,
traditional design methods do not permit the elimination of such unviable
candidate
configurations without the expense of a detailed ray matrix analysis.
[00084] FIG. 17 illustrates how a PDWS 1700, such as PDWS 300 or PDWS 1400,
can be
used to measure both surfaces of a lens 17, for example, a contact lens or an
intraocular lens.
Light from a light source 1710 is passed through a beamsplitter 1720 to lens
17. Reflections
are produced from both surfaces of lens 17 and pass back through beamsplitter
to the PDWS
1700 which has sample planes located about the focal positions of the light
reflected from the
two lens surfaces. Here, the advantages of PDWS 1700 can be seen. For example,
if a
SHWS were employed in this application, the multiple reflections from the
surfaces of lens
17 would generate multiple focal spots from its lenslet array that could
confuse the processor
associated with a SHWS. In contrast, PDWS 1700 can easily distinguish between
the two
reflected wavefronts, and therefor both surfaces of lens 17 can be
characterized. In this
example, the wave reflected from each surface will focus at different
distances from the lens;
it is obvious that by suitably placing sufficient PDWS sample planes near
these foci,
sufficient data can be made available to a Gerchberg-Saxton phase retrieval
algorithm to
determine the wavefront from each surface and hence the optical effect of each
surface.
More than two sample planes may be required in such multi wavefront
applications and their
number and locations may be expected to affect the accuracy of the phase
retrieval.
Generally the greater the number of sample planes the greater the accuracy of
the phase
retrieval; likewise, judiciously placing the sample planes around the
locations where the
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intensities contributed at those planes by the various wavefronts are most
disparate will lead
to the greatest accuracy in retrieving each component in the multi wavefront.
[00085] FIGs. 18A-C illustrate the use of a PDWS in an ophthalmic measurement
application. FIGs. 18A-C show ray trace results from a non-paraxial analysis.
The PDWS
configuration illustrated in FIGs. 18A-C has a 300 mm pupil plane (standoff)
distance, and
the camera has 300 pixels across a width of 6 mm. Other parameters include:
first lens with f
= 100 mm and 25 mm dial.; second lens with fl = 300 mm and 38 mm dia.; DOE
with fg =
500 mm and 9 [tm center grating period. FIG. 18A illustrates a case where +10
diopters of
ophthalmic correction are required; FIG. 18B illustrates a case where 0
diopters of
ophthalmic correction are required; and FIG. 18C illustrates a case where -10
diopters of
ophthalmic correction are required. The ray trace analysis shown in FIGs. 18A-
C shows that
light rays are fully transmitted to the camera in this arrangement for beams
within the range +
diopters of defocus; for this reason, this configuration is suitable to
acquire the data
necessary to analyze beams with this wide range of defocus. Indeed even larger
ranges may
be possible by increasing the diameter of the second lens. The fact that the
second lens is
quite nearly filled by the rays in the +10 diopter case suggests that some non
paraxial
behavior may be expected in this limit. The detailed ray trace analysis of
such a system
employing realistic commercially available lenses shows that the non-paraxial
behavior of
the system magnification departs from ideal by only about 1.3% at the extremes
of the
dynamic range, well within the acceptable tolerance for an ophthalmic
aberrometer.
[00086] FIG. 19 illustrates a block diagram of one embodiment of an ophthalmic
aberrometer 1900 that includes a PDWS 1910, which for example can be PDWS 300
or
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PDWS 1400. Ophthalmic aberrometer 1900 also includes a light source 1920, an
optical
system 1930, and a processor 1950.
1000871 While preferred cmbodimcnts arc disclosed herein, many variations will
occur to
one of ordinary skill in the art. The scope of the claims should not be
limited by the
preferred embodiments or the examples but should be given the broadest
interpretation
consistent with the description as a whole.
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