Freeform surface imaging spectrometer system
A freeform surface imaging spectrometer system including a primary mirror, a secondary mirror, a tertiary mirror, and a detector is provided. The secondary mirror is a grating having a freeform surface shape, and the grating having the freeform surface shape is obtained by intersecting a set of equally spaced parallel planes with a freeform surface. A plurality of feature rays exiting from a light source is successively reflected by the primary mirror, the secondary mirror and the tertiary mirror to form an image on an image sensor. A reflective surface of each of the primary mirror, the tertiary mirror surface and the tertiary mirror is an xy polynomial freeform surface.
1. A freeform surface imaging spectrometer system comprising:
a primary mirror, wherein a global three-dimensional rectangular coordinate system (X 0 , Y 0 , Z 0 ) is established in a space wherein a slit of the freeform surface imaging spectrometer system is located, an origin of the global three-dimensional rectangular coordinate system (X 0 , Y 0 , Z 0 ) is an object point of a central field of view on the slit, and the global three-dimensional rectangular coordinate system (X 0 , Y 0 , Z 0 ) comprises an X-axis, a Y-axis, and a Z-axis; and a first three-dimensional rectangular coordinates system (X, Y, Z) is established, a first origin of the first three-dimensional rectangular coordinates system (X, Y, Z) is a vertex of the primary mirror, and the first three-dimensional rectangular coordinates system (X, Y, Z) is obtained by moving the global three-dimensional rectangular coordinate system (X 0 , Y 0 , Z 0 ) along a Z-axis positive direction of the Z-axis and a Y-axis positive direction of the Y-axis; and a reflective surface of the primary mirror in the first three-dimensional rectangular coordinates system (X, Y, Z) is an xy polynomial freeform surface;
a secondary mirror being as an aperture stop, wherein the secondary mirror is a grating a grating pitch of the grating is ranged from about 0.004 millimeters to about 0.010 millimeters; and a second three-dimensional rectangular coordinates system (X′, Y′, Z′) is established, wherein a second origin of the second three-dimensional rectangular coordinates system (X′, Y′, Z′) is a vertex of the secondary mirror, and the second three-dimensional rectangular coordinates system (X′, Y′, Z′) is obtained by moving the global three-dimensional rectangular coordinate system (X 0 , Y 0 , Z 0 ) along the positive direction of the Z-axis and a negative direction of the Y-axis; and a reflective surface of the secondary mirror in the second three-dimensional rectangular coordinates system (X′, Y′, Z′) is an x′y′ polynomial freeform surface;
a tertiary mirror, wherein a third three-dimensional rectangular coordinates system (X″, Y″, Z″) is established, and the third three-dimensional rectangular coordinates system (X″, Y″, Z″) is obtained by moving the global three-dimensional rectangular coordinate system (X 0 , Y 0 , Z 0 ) along the positive direction of the Z-axis and the positive direction of the Y-axis; and a reflective surface of the tertiary mirror in the third three-dimensional rectangular coordinates system (X″, Y″, Z″) is an x″y″ polynomial freeform surface; and
a detector, wherein a plurality of feature rays exiting from a light source is successively reflected by the primary mirror, the secondary mirror and the tertiary mirror and forms an image on the detector,
wherein a spectral resolution of the freeform surface imaging spectrometer system is equal to or more than 2 nanometers.
2. The system as claimed in claim 1 , wherein the xy polynomial freeform surface is an sixth-order polynomial freeform surface of xy without odd items of x; and an equation of the sixth-order polynomial freeform surface of xy is:
z
(
x
,
y
)
=
c
(
x
2
+
y
2
)
1
+
1
-
(
1
+
k
)
c
2
(
x
2
+
y
2
)
+
A
2
y
+
A
3
x
2
+
A
5
y
2
+
A
7
x
2
y
+
A
9
y
3
+
A
10
x
4
+
A
12
x
2
y
2
+
A
14
y
4
+
A
16
x
4
y
+
A
18
x
2
y
3
+
A
20
y
5
+
A
21
x
6
+
A
23
x
4
y
2
+
A
25
x
2
y
4
+
A
27
y
6
,
wherein z represents surface sag, c represents surface curvature, k represents conic constant, while A i represents the ith term coefficient.
3. The system as claimed in claim 2 , wherein c=1/−145.418275620109, k=0.624958846203866, A 2 =−0.0163788177487878, A 3 =−0.000295233410827397, A 5 =−0.000210300869303387, A 7 =4.66485198772164e-006, A 9 =−4.21385063387942e-007, A 10 =1.36627658326689e-008, A 12 =−3.02182908797448e-009, A 14 =1.3081464012072e-008, A 16 =1.97565367144866e-010, A 18 =3.16770638046019e-010, A 20 =−3.79206991893053e-011, A 21 =1.17652940989923e-012, A 23 =1.44167031123941e-012, A 25 =4.98199759406961e-013, and A 27 =7.80274289683532e-013.
4. The system as claimed in claim 1 , wherein the x′y′ polynomial freeform surface is an sixth-order polynomial freeform surface of x′y′ without odd items of x′; and an equation of the sixth-order polynomial freeform surface of x′y′ is:
z
′
(
x
′
,
y
′
)
=
c
′
(
x
′2
+
y
′2
)
1
+
1
-
(
1
+
k
′
)
c
′2
(
x
′2
+
y
′2
)
+
A
2
′
y
′
+
A
3
′
x
′2
+
A
5
′
y
′2
+
A
7
′
x
′2
y
′
+
A
9
′
y
′3
+
A
10
′
x
′4
+
A
12
′
x
′2
y
′2
+
A
14
′
y
′4
+
A
16
′
x
′4
y
′
+
A
18
′
x
′2
y
′3
+
A
20
′
y
′5
+
A
21
′
x
′6
+
A
23
′
x
′4
y
2
′
+
A
25
′
x
2
′
y
′4
+
A
27
′
y
′6
,
wherein z′ represents surface sag, c′ represents surface curvature, k′ represents conic constant, while A i ′ represents the ith term coefficient.
5. The system as claimed in claim 4 , wherein c′=1/−74.787222409682, k′=0.197915885616308, A 2 ′=0.219139600693203, A 3 ′=−0.0015668216458602, A 5 ′=−0.000113148729752593, A 7 ′=0.000110773994138095, A 9 ′=4.64610222017242e-006, A 10 ′=−1.41006542846292e-006, A 12 ′=−1.8434653293837e-006, A 14 ′=−1.53743088794955e-007, A 16 ′=6.56200181475036e-008, A 18 ′=3.31077127158278e-008, A 20 ′=−2.98732023163831e-009, A 21 ′=−8.1605169185165e-010, A 23 ′=−1.92331290018676e-009, A 25 ′=−6.01563974506452e-010, and A 27 ′=−1.67959645045492e-010.
6. The system as claimed in claim 1 , wherein the x″y″ polynomial freeform surface is an sixth-order polynomial freeform surface of x″y″ without odd items of x″; and an equation of the sixth-order polynomial freeform surface of x″y″ is:
z
″
(
x
″
,
y
″
)
=
c
″
(
x
″2
+
y
″2
)
1
+
1
-
(
1
+
k
″
)
c
″2
(
x
″2
+
y
″2
)
+
A
2
″
y
″
+
A
3
″
x
″2
+
A
5
″
y
″2
+
A
7
″
x
″2
y
″
+
A
9
″
y
″3
+
A
10
″
x
″4
+
A
12
″
x
″2
y
″2
+
A
14
″
y
″4
+
A
16
″
x
″4
y
″
+
A
18
″
x
″2
y
″3
+
A
20
″
y
″5
+
A
21
″
x
″6
+
A
23
″
x
″4
y
″2
+
A
25
″
x
″2
y
″4
+
A
27
″
y
″6
,
wherein z″ represents surface sag, c″ represents surface curvature, k″ represents conic constant, while A i ″ represents the ith term coefficient.
7. The system as claimed in claim 6 , wherein c″=1/−125.336259302399, k″=0.730866579643393, A 2 ″=−0.013535502861499, A 3 ″=−0.000248904316215348, A 5 ″=−7.0630777168571e-005, A 7 ″=−1.67104431028359e-006, A 9 ″=−2.59911394183642e-006, A 10 ″=1.53893531673435e-008, A 12 ″=5.35895831746943 e-008, A 14 ″=−2.03909100495527e-010, A 16 ″=3.52133563125681e-008, A 18 ″=−3.41817171152247e-010, A 20 ″=−1.52203050072224e-010, A 21 ″=1.84098557538911e-012, A 23 ″=8.40400504710118e-012, A 25 ″=7.32052575487887e-012, and A 27 ″=1.76461076891148e-012.
8. The system as claimed in claim 1 , wherein an F-number of the freeform surface imaging spectrometer system is less than or equal to 3.0.
9. The system as claimed in claim 1 , wherein a numerical aperture of the freeform surface imaging spectrometer system ranges from about 0.1 to about 0.2.
10. The system as claimed in claim 1 , wherein a slit length of the freeform surface imaging spectrometer system is ranged from about 8 millimeters to about 12 millimeters.
11. The system as claimed in claim 1 , wherein a slit width of the freeform surface imaging spectrometer system ranges from about 6 micrometers to about 10 micrometers.
12. The system as claimed in claim 1 , wherein the first three-dimensional rectangular coordinates system (X, Y, Z) is obtained by moving the global three-dimensional rectangular coordinate system (X 0 , Y 0 , Z 0 ) for about 0.375 mm along the Y-axis in positive direction, and then moving for about 130.943 mm along the Z-axis in positive direction, and then rotating in a counterclockwise direction for about 5.188 degrees with the X axis being a rotation axis.
13. The system as claimed in claim 1 , wherein the second three-dimensional rectangular coordinates system (X′, Y′, Z′) is obtained by moving the global three-dimensional rectangular coordinate system (X 0 , Y 0 , Z 0 ) for about 18.001 mm along the Y-axis in negative direction, and then moving for about 46.046 mm along the Z-axis in positive direction, and then rotating in a counterclockwise direction for about 43.336 degrees with the X axis being a rotation axis.
14. The system as claimed in claim 1 , wherein the third three-dimensional rectangular coordinates system (X″, Y″, Z″) is obtained by moving the global three-dimensional rectangular coordinate system (X 0 , Y 0 , Z 0 ) for about 93.370 mm along the Z-axis in positive direction, and then moving for about 44.689 mm along the Y-axis in positive direction, and then rotating in a counterclockwise direction for about 47.862 degrees with the X axis being the rotation axis.
15. The system as claimed in claim 1 , wherein a distance between the first origin and the origin of the global three-dimensional rectangular coordinate system (X 0 , Y 0 , Z 0 ) is about 130.997 mm.
16. The system as claimed in claim 1 , wherein a distance between the second origin and the origin of the global three-dimensional rectangular coordinate system (X 0 , Y 0 , Z 0 ) is about 49.442 mm.