Precision freeform structuring for the fabrication of coded lenses
A system for fabricating coded lenses includes a cutting tool configured to controllably cut a workpiece at a specified position-dependent depth while traversing a surface of the workpiece along a specified two-dimensional path. A signal generator is operative to generate a signal for controlling fabrication of a coded lens from the workpiece. A vibration tool is operative to ultrasonically vibrate the cutting tool for cutting of gratings on the workpiece.
1 . A system for fabricating coded lenses comprising:
a cutting tool configured to controllably cut a workpiece at a specified position-dependent depth while traversing a surface of the workpiece along a specified two-dimensional path;
a signal generator operative to generate a signal for controlling fabrication of a coded lens from the workpiece; and
a vibration tool operative to ultrasonically vibrate the cutting tool for cutting of gratings on the workpiece, wherein the cutting tool is configured to machine a freeform surface and fabricate a variable grating on the machined freeform surface; and
a processor configured to control the cutting tool and the vibration tool for machining a freeform lens from the workpiece, wherein computing the three-dimensional freeform surface is based on a Chebyshev polynomial
Z
=
f
z
(
X
,
Y
)
=
-
(
c
(
X
2
+
Y
2
)
1
+
1
+
c
2
(
X
2
+
Y
2
)
+
∑
i
=
0
I
∑
j
=
0
J
C
ij
T
i
(
X
L
x
)
T
j
(
X
L
y
)
+
Z
0
T
i
(
x
)
=
cos
(
i
arccos
x
)
,
where x, y, z are global coordinates, X, Y, Z are workpiece coordinates, ρ, θ are workpiece polar coordinates, Z=f z (X,Y) is a freeform surface function, Z 0 is a reference depth, T i (x) is a i th order Chebyshev polynomials at x, C ij is a 2-D Chebyshev polynomials coefficient, i, j are a order of Chebyshev polynomials, I, J are a maximum order of Chebyshev polynomials, c is a curvature of toric term.
2 . The system of claim 1 , wherein the cutting tool is configured to rotate to cut the workpiece in directions that are perpendicular to local gradients on the workpiece.
3 . The system of claim 1 , further comprising a vacuum to remove cut material from the cutting tool while the cutting tool is cutting the workpiece.
4 . The system of claim 1 , further comprising a mechanism to feed the workpiece linearly along an axis relative to the cutting tool while the cutting tool cuts the workpiece.
5 . The system of claim 1 , further comprising a spindle operative to hold the workpiece and turn the workpiece about a central axis to facilitate the cutting tool to cut the workpiece while the workpiece is turning.
6 . The system of claim 1 , wherein the vibration tool includes an elliptical vibration tool.
7 . A method for fabricating coded lenses comprising:
generating a signal for controlling fabrication of a coded lens from a workpiece;
causing a cutting tool to traverse a surface of the workpiece along a specified two-dimensional path, based on the signal;
controllably cutting a workpiece at a specified position-dependent depth while traversing the surface of the workpiece along the specified two-dimensional path, based on the signal; and
ultrasonically vibrating the cutting tool for cutting gratings on the workpiece; and
computing a three-dimensional freeform surface for machining a freeform lens from the workpiece, wherein computing the three-dimensional freeform surface is based on a Chebyshev polynomial
Z
=
f
z
(
X
,
Y
)
=
-
(
c
(
X
2
+
Y
2
)
1
+
1
-
c
2
(
X
2
+
Y
2
)
+
∑
i
=
0
I
∑
j
=
0
J
C
ij
T
i
(
X
L
x
)
T
j
(
X
L
y
)
+
Z
0
T
i
(
x
)
=
cos
(
i
arccos
x
)
where x, y, z are global coordinates, X, Y, Z are workpiece coordinates, ρ, θ are workpiece polar coordinates, Z=f z (X,Y) is a freeform surface function, Z 0 is a reference depth, T i (x) is a i th order Chebyshev polynomials at x, C ij is a 2-D Chebyshev polynomials coefficient, i, j are a order of Chebyshev polynomials, I, J are a maximum order of Chebyshev polynomials, c is a curvature of toric term.
8 . The method of claim 7 , further comprising causing the cutting tool to traverse the surface of the workpiece based on the Chebyshev polynomial to machine a freeform surface.
9 . The method of claim 7 , wherein causing the cutting tool to traverse the surface of the workpiece along the specified two-dimensional path comprises feeding the workpiece along a linear direction at a variable rate.
10 . The method of claim 7 , wherein causing the cutting tool to traverse the surface of the workpiece along the specified two-dimensional path comprises causing the cutting tool to traverse the surface of the workpiece along quasi-spiral tool paths computed according to
M
n
≈
h
/
Δ
r
min
k
=
1
,
…
,
K
∇
f
(
X
k
,
n
,
Y
k
,
n
)
,
where at a revolution m from 1 to M n , the quasi-spiral tool path is computed by an interpolation, where a k th point on a m th revolution is labelled as (X k,m , Y k,m ),
where
X
k
,
m
=
m
(
k
-
1
)
M
n
K
(
X
k
,
n
+
1
-
X
k
,
n
)
+
X
k
,
n
,
Y
k
,
m
=
m
(
k
-
1
)
M
n
K
(
Y
k
,
n
+
1
-
Y
k
,
n
)
+
Y
k
,
n
,
Z
k
,
m
=
f
z
(
X
k
.
m
,
Y
k
,
m
)
-
(
n
-
1
)
h
+
e
k
(
X
k
,
m
,
Y
k
,
m
)
,
where e h (X k,m , Y k,m ) is a tool compensation term.
11 . The method of claim 7 , wherein the specified two-dimensional path is computed according to
[
X
k
,
m
Y
k
,
m
∇
x
f
(
X
k
,
m
,
Y
k
,
m
)
∇
y
f
(
X
k
,
m
,
Y
k
,
m
)
]
=
[
cos
(
φ
k
,
m
)
sin
(
φ
k
,
m
)
0
0
-
sin
(
φ
k
,
m
)
cos
(
φ
k
,
m
)
0
0
0
0
cos
(
φ
k
,
m
)
sin
(
φ
k
,
m
)
0
0
-
sin
(
φ
k
,
m
)
cos
(
φ
k
,
m
)
]
[
x
k
,
m
y
k
,
m
-
1
0
]
,
∂
f
(
X
,
Y
)
∂
X
=
cX
1
-
c
2
(
X
2
+
Y
2
)
+
∑
i
=
0
i
∑
j
=
0
j
C
ij
i
sin
(
i
arc
cos
(
X
L
x
)
)
1
-
(
X
L
x
)
2
cos
(
j
arc
cos
(
Y
L
y
)
)
∂
f
(
X
,
Y
)
∂
Y
=
cX
1
-
c
2
(
X
2
+
Y
2
)
+
∑
i
=
0
i
∑
j
=
0
j
C
ij
cos
(
i
arc
cos
(
X
L
x
)
)
j
sin
(
j
arc
cos
(
Y
L
y
)
)
1
-
(
Y
L
y
)
2
such that a cutting tool's trajectory in a workpiece coordinate system (X, Y, Z) is on a desired quasi-spiral, and a cutting direction is perpendicular to a local gradient.
12 . The method of claim 7 , further comprising compensating for a deviation of a cutting tool's actual cutting point from the desired cutting point on the workpiece by computing e h according to
e
h
=
R
tan
(
β
)
tan
(
β
2
)
β
=
arc
tan
(
∇
f
z
(
X
,
Y
)
)
and adjusting the cutting tool's actual cutting point based on e h .
13 . A non-transitory machine-readable storage medium having instructions stored thereon for causing a processor to execute the method:
generating a signal for controlling fabrication of a coded lens from a workpiece;
causing a cutting tool to traverse a surface of the workpiece along a specified two-dimensional path, based on the signal;
controllably cutting a workpiece at a specified position-dependent depth while traversing the surface of the workpiece along the specified two-dimensional path, based on the signal;
ultrasonically vibrating the cutting tool for cutting gratings on the workpiece; and
computing a three-dimensional freeform surface for machining a freeform lens from the workpiece, wherein computing the three-dimensional freeform surface is based on a Chebyshev polynomial
Z
=
f
z
(
X
,
Y
)
=
-
(
c
(
X
2
+
Y
2
)
1
+
1
-
c
2
(
X
2
+
Y
2
)
+
∑
i
=
0
I
∑
j
=
0
J
C
ij
T
i
(
X
L
x
)
T
j
(
Y
L
y
)
)
+
Z
0
,
T
i
(
x
)
=
cos
(
i
arc
cos
x
)
where x, y, z are global coordinates, X, Y, Z are workpiece coordinates, ρ, θ are workpiece polar coordinates, Z=f z (X,Y) is a freeform surface function, Z 0 is a reference depth, T i (x) is a i th order Chebyshev polynomials at x, C ij is a 2-D Chebyshev polynomials coefficient, i, j are a order of Chebyshev polynomials, I, J are a maximum order of Chebyshev polynomials, c is a curvature of toric term.
14 . The medium of claim 13 , wherein the instructions for causing the cutting tool to traverse the surface of the workpiece along the specified two-dimensional path further comprise instructions for feeding the workpiece along a linear direction at a variable rate.
15 . The medium of claim 13 , wherein the instructions cause the processor to further execute the method operations:
causing the cutting tool to traverse the surface of the workpiece along quasi-spiral tool paths computed according to
M
n
≈
h
/
Δ
r
min
k
=
1
,
…
,
K
∇
f
(
X
k
,
n
,
Y
k
,
n
)
,
where at a revolution m from 1 to M n , the quasi-spiral tool path is computed by a interpolation, where a k th point on a m th revolution is labelled as (X k,m , Y k,m ), where
X
k
,
m
=
m
(
k
-
1
)
M
n
K
(
X
k
,
n
+
1
-
X
k
,
n
)
+
X
k
,
n
,
Y
k
,
m
=
m
(
k
-
1
)
M
n
K
(
Y
k
,
n
+
1
-
Y
k
,
n
)
+
Y
k
,
n
,
Z
k
,
m
=
f
z
(
X
k
,
m
,
Y
k
,
m
)
-
(
n
-
1
)
h
+
e
k
(
X
k
,
m
,
Y
k
,
m
)
,
where e h (X k,m , Y k,m ) is a tool compensation term.
16 . The medium of claim 13 , wherein the instructions cause the processor to further execute the method operations:
causing the cutting tool to traverse the surface of the workpiece along the specified two-dimensional path computed according to
[
X
k
,
m
Y
k
,
m
∇
x
f
(
X
k
,
m
,
Y
k
,
m
)
∇
y
f
(
X
k
,
m
,
Y
k
,
m
)
]
=
[
cos
(
φ
k
,
m
)
sin
(
φ
k
,
m
)
0
0
-
sin
(
φ
k
,
m
)
cos
(
φ
k
,
m
)
0
0
0
0
cos
(
φ
k
,
m
)
sin
(
φ
k
,
m
)
0
0
-
sin
(
φ
k
,
m
)
cos
(
φ
k
,
m
)
]
[
x
k
,
m
y
k
,
m
-
1
0
]
,
∂
f
(
X
,
Y
)
∂
X
=
cX
1
-
c
2
(
X
2
+
Y
2
)
+
∑
i
=
0
i
∑
j
=
0
j
C
ij
i
sin
(
i
arc
cos
(
X
L
x
)
)
1
-
(
X
L
x
)
2
cos
(
j
arc
cos
(
Y
L
y
)
)
∂
f
(
X
,
Y
)
∂
Y
=
cX
1
-
c
2
(
X
2
+
Y
2
)
+
∑
i
=
0
i
∑
j
=
0
j
C
ij
cos
(
i
arc
cos
(
X
L
x
)
)
j
sin
(
j
arc
cos
(
Y
L
y
)
)
1
-
(
Y
L
y
)
2
such that a cutting tool's trajectory in a workpiece coordinate system (X, Y, Z) is on a desired quasi-spiral, and a cutting direction is perpendicular to a local gradient.
17 . The medium of claim 13 , wherein the instructions cause the processor to further execute the method operations:
compensating for a deviation of a cutting tool's actual cutting point from the desired cutting point on the workpiece by computing e h according to
e
h
=
R
tan
(
β
)
tan
(
β
2
)
β
=
arc
tan
(
∇
f
z
(
X
,
Y
)
)
and adjusting the cutting tool's actual cutting point based on e h .