US 2006/0039074 A1 · Optical imaging device designs for solar-powered flight and powergeneration

A method of designing a two-mirror high numerical aperture imaging device comprising the steps of:

(a) determining the positioning of each consecutive point of a cross-section through the x-axis of a first mirror and a second mirror, iteratively for a cross-section in the plane Z=0, where the x and y coordinates of each successive point on the two mirrors in the cross-section are M1(t)≡(m1,x(t),m1,y(t),0) (for the first mirror) and M2(t)≡(m2,x(t),m2,y(t),0) (for the second), t being an iteration counter;
where the functions M0(t)≡(m0,x(t), m0,y(t),0)≡(f,0,0) and M3 (t)≡(m3,x(t), m3,y(t),0)≡(0,0,0) are set to define the centres of the object and image planes respectively;
(b) Calculating the angle di(t) (for i=0, . . . , 2) that a ray from Mi(t) to Mi+1(t) makes to the x-axis, i.e.
d
i
⁡
(
t
)
=
arctan
⁡
(
m
i
+
1
,
y
⁡
(
t
)
-
m
i
,
y
⁡
(
t
)
m
i
+
1
,
x
⁡
(
t
)
-
m
i
,
x
⁡
(
t
)
)
(c) Calculating the distance pi(t) (for i=0, . . . , 2) between Mi(t) (and Mi+1(t), i.e.
pi(t)=√{square root over ((mi+1,x(t)−mi,x(t))2+(mi+1,y(t)−mi,y(t))2)}
(d) Calculating the angle ai(t) that a tangent to the i'th surface makes to the x-axis, taking a0(t) and a3(t) to be the angles that the object and image planes make to the x-axis (being 90° for all t for the surfaces to be rotationally symmetric about the x-axis), i.e. for reflection (for i=1 and 2):
a
i
⁡
(
t
)
=
d
i
⁡
(
t
)
2
+
d
i
-
1
⁡
(
t
)
2
(e) Choosing the values of mi,x(0) and mi,y(0) (for i=1 and 2) so that the resulting mirror layout satisfies the sine criterion, and wherein for a far away source (say along the negative x-axis, say f=−109) a scale parameter, b may be set equal to unity (without loss of generality, such that B=b/p0(0)=1/p0(0)), and wherein the sine criterion is then satisfied if:
m
1
,
y
⁡
(
0
)
=
±
m
2
,
y
⁡
(
0
)
(
m
2
,
x
⁡
(
0
)
2
+
m
2
,
y
⁡
(
0
)
2
)
1
/
2
(f) Iterating either from shallower angles to more oblique angles, or vice versa (switching between the two being achieved by using as seed values later iterated results and reversing the sign of the iteration step size h), such that if the iteration is from highly oblique angles then set m2,x(0) equal to a small number close to zero, say 10−9, and choose m1,x(0)=q1 and m2,y(0)=q2, where q1 and q2 are arbitrary real numbers (positive or negative), but with the choice of signs of q1, q2 and h constrained so as not to have simultaneously the sign of q1 negative, the sign of q2 positive and the sign of h positive, then m1,y(0) will need to be ±1 for the initial parameters to satisfy the sine criterion, and without loss of generality it is possible to choose m1,y(0) to be −1;
(g) Choosing the sign Z so that the sine criterion remains satisfied as t changes,
(h) Updating the values of Mi(t) as follows (for a small value of h):
M
i
⁡
(
t
+
1
)
≡
(
m
i
,
x
⁡
(
t
+
1
)
m
i
,
y
⁡
(
t
+
1
)
)
=
M
i
⁡
(
t
)
+
w
i
⁡
(
cos
⁡
(
a
i
⁡
(
t
)
)
sin
⁡
(
a
i
⁡
(
t
)
)
)
⁢
h
where ri-1(t)=sin(ai-1(t)−di-1(t)) si(t)=sin(ai(t)−di-1(t)) and where
w
2
=
p
1
⁢
r
0
p
0
⁢
s
2
⁢
⁢
w
1
=
-
ZB
⁢
p
1
⁢
s
3
p
2
⁢
r
1
(i) Ending the iteration no later than when light rays cease to be able to pass freely through the mirror arrangement, once it has been rotated as in (j);
(j) Rotating the curves produced above around the x-axis to define the complete, three-dimensional mirror surfaces.

222 more elements

A high numerical aperture imaging device comprising:

a first mirror and a second mirror, wherein the positioning of each consecutive point of a cross-section through the x-axis of a first mirror and a second mirror is determined iteratively for a cross-section in the plane z=0, where the x and y coordinates of each successive point on the two mirrors in the cross-section are M1(t)≡(m1,x(t), m1,y(t),0) (for the first mirror) and M2 (t)≡(m2,x(t), m2,y(y),0) (for the second), t being an iteration counter;
where the functions M0(t)≡(m0,x(t), m0,y(t),0)≡(f,0,0) and M3(t)≡(m3,x(t), m3,y(t),0)≡(0,0,0) are set to define the centres of the object and image planes respectively;
wherein the angle di(t) (for i=0, . . . , 2) that a ray from Mi(t) to Mi+1(t) makes to the x-axis is given by:
d
i
⁡
(
t
)
=
arctan
⁡
(
m
i
+
1
,
y
⁡
(
t
)
-
m
i
,
y
⁡
(
t
)
m
i
+
1
,
x
⁡
(
t
)
-
m
i
,
x
⁡
(
t
)
)
wherein the distance pi(t) (for i=0, . . . , 2) between Mi(t) and Mi+1(t) is given by:
pi(t)=√{square root over ((mi+1,x(t)−mi,x(t))2+(mi+1,y(t)−mi,y(t))2)}
wherein the angle ai(t) that a tangent to the i+th surface makes to the x-axis, taking a0(t) and a3(t) to be the angles that the object and image planes make to the x-axis (being 90° for all t for the surfaces to be rotationally symmetric about the x-axis), i.e. for reflection (for i=1 and 2), is given by:
a
i
⁡
(
t
)
=
d
i
⁡
(
t
)
2
+
d
i
-
1
⁡
(
t
)
2
wherein the values of mi,x(0) and mi,y(0) (for i=1 and 2) are chosen so that the resulting mirror layout satisfies the sine criterion;
wherein for a far away source (say along the negative x-axis, say f=−109) a scale parameter, b may be set equal to unity (without loss of generality, such that B=b/p0(0)=1/p0(0));
wherein the sine criterion is then satisfied if:
m
1
,
y
⁡
(
0
)
=
±
m
2
,
y
⁡
(
0
)
(
m
2
,
x
⁡
(
0
)
2
+
m
2
,
y
⁡
(
0
)
2
)
1
/
2
wherein iterating either from shallower angles to more oblique angles, or vice versa (switching between the two being achieved by using as seed values later iterated results and reversing the sign of the iteration step size h), such that if the iteration is from highly oblique angles, then set m2,x(0) equal to a small number close to zero, say 10−9, and choose m1,x(0)=q1 and m2,y(0)=q2, where q1 and q2 are arbitrary real numbers (positive or negative), but with the choice of signs of q1, q2 and h constrained so as not to have simultaneously the sign of q1 negative, the sign of q2 positive and the sign of h positive, then m1,y(0) will need to be ±1 for the initial parameters to satisfy the sine criterion;
and wherein without loss of generality it is possible to choose m1,y(0) to be −1;
wherein the sign Z is chosen so that the sine criterion remains satisfied as t changes;
wherein the values of Mi(t) are updated as follows (for a small value of h);
M
i
⁡
(
t
+
1
)
≡
(
m
i
,
x
⁡
(
t
+
1
)
m
i
,
y
⁡
(
t
+
1
)
)
=
M
i
⁡
(
t
)
+
w
i
⁡
(
cos
⁢
⁢
(
a
i
⁡
(
t
)
)
sin
⁡
(
a
i
⁡
(
t
)
)
)
⁢
h
where ri-1(t)=sin(ai-1(t)−di-1(t)) si(t)=sin(ai(t)−di-1(t)) and where
w
2
=
p
1
⁢
r
0
p
0
⁢
s
2
⁢
⁢
w
1
=
-
ZB
⁢
p
1
⁢
s
3
p
2
⁢
r
1
wherein the iteration is ended no later than when light rays cease to be able to pass freely through the mirror arrangement, once it has been rotated as in step (j);
and
wherein the curves produced above are rotated around the x-axis to define the complete, three-dimensional mirror surfaces.

227 more elements

Optical imaging device designs for solar-powered flight and powergeneration

Life of this patent

Optical imaging device designs for solar-powered flight and powergeneration

Life of this patent

Abstract

Claims23