Method of wavenumber linearity dispersion optical system and imaging spectrometer
The invention discloses a design method of a wavenumber linearity dispersion optical system and an imaging spectrometer, including: building an optical system including a grating, a prism and an objective lens that are sequentially arranged, the grating adjoins the prism; defining a linearity evaluation coefficient RMS; assigning a minimum value to the linearity evaluation coefficient RMS through adjustment to the vertex angle of the prism, when the linearity evaluation coefficient RMS is at minimum, the vertex angle of the prism being α 1 ; acquiring compensations for distortion and longitudinal chromatic aberration of the objective lens based on the interval between equal-difference wavenumbers on the image plane when the vertex angle of the prism is α 1 ; and optimizing the objective lens based on the compensations for distortion and longitudinal chromatic aberration of the objective lens to obtain an optimized optical system. Higher wavenumber linearity can be achieved through objective-lens-aberration compensated wavenumber linearity.
1 . A design method of a wavenumber linearity dispersion optical system, comprising steps of:
S 1 : building an optical system including a grating, a prism and an objective lens that are sequentially arranged, the grating adjoining the prism, wherein collimated light is split through the grating and the prism into rays of different wavenumbers, and the rays of different wavenumbers enter the objective lens at different angles and image on an image plane;
S 2 : defining a linearity evaluation coefficient RMS, which is the root-mean-square error of intervals between equal-difference wavenumbers in an operating band on the image plane;
S 3 : assuming a length of the image plane Y as a constant value, assigning a minimum value to the linearity evaluation coefficient RMS through adjustment to the vertex angle of the prism α, when the linearity evaluation coefficient RMS is at minimum, the vertex angle of the prism is α 1 ;
S 4 : acquiring compensations for distortion and longitudinal chromatic aberration of the objective lens based on the interval between the equal-difference wavenumbers in the image plane when the vertex angle of the prism is α 1 ; and
S 5 : optimizing the objective lens based on the compensations for distortion and longitudinal chromatic aberration of the objective lens to obtain an optimized optical system.
2 . The design method of a wavenumber linearity dispersion optical system of claim 1 , wherein between the steps S 4 and S 5 the method further comprises designing an objective lens in which negative distortion and longitudinal chromatic aberration can be introduced.
3 . The design method of a wavenumber linearity dispersion optical system of claim 2 , wherein the objective lens includes a first positive lens, a first negative lens, a second positive lens and a third positive lens that are sequentially arranged, the incident height and incident angle of an off-axis view field chief ray on the third positive lens are increased to produce negative spherical aberration in order to introduce negative distortion.
4 . The design method of a wavenumber linearity dispersion optical system of claim 3 , wherein the second positive lens is located at a first side of the third positive lens, and a second negative lens is provided at a second side of the third positive lens to correct the field curvature.
5 . The design method of a wavenumber linearity dispersion optical system of claim 4 , wherein the first positive lens, the first negative lens, the second positive lens, the third positive lens and the second negative lens are made of the same material to introduce longitudinal chromatic aberration.
6 . The design method of a wavenumber linearity dispersion optical system of claim 5 , wherein the first positive lens, the first negative lens, the second positive lens, the third positive lens and the second negative lens have a refractive index in the range of 1.5 to 2.3.
7 . The design method of a wavenumber linearity dispersion optical system of claim 5 , wherein the step S 5 comprises optimizing the optical system by changing the curvature of the objective lens, the interval between adjacent lenses, the thickness of the lens, and the material of the lens.
8 . The design method of a wavenumber linearity dispersion optical system of claim 1 , wherein the step S 2 comprises steps of:
S 21 : selecting n equal-difference wavenumbers from the operating band;
S 22 : assuming the incident angle of a collimated ray on the grating as θ in , the diffraction angle as θ d , an included angle between the grating and the prism as β, a vertex angle of the prism as α, incident angles of the ray on the front and back surfaces of the prism respectively as θ 1 and θ 3 and the corresponding exit angles respectively as θ 2 and θ, the refractive index of the prism as n(λ), and an exit angle of the center wavenumber
k
n
+
1
2
on the back surface of the prism as
θ
k
n
+
1
2
,
with its exit direction as the optical axis direction of the objective lens, and based on the geometrical relationship, the grating equation and the law of refraction, obtaining:
sin
θ
2
=
sin
[
β
+
arcsin
(
λ
d
-
sin
θ
in
)
]
n
(
λ
)
,
(
1
)
sin
θ
=
n
(
λ
)
sin
θ
3
,
(2)
where d is the grating constant, λ=2π/k is the wavelength of the light, β=θ 1 −θ d , and θ 3 =α−θ 2 ;
S 23 : combining equation (1) with equation (2) and obtaining the exit angle of the ray on the back surface of the prism as:
θ
=
arcsin
{
n
(
λ
)
·
sin
{
α
-
arcsin
{
sin
{
arcsin
(
λ
d
-
sin
θ
in
)
+
β
}
n
(
λ
)
}
}
}
;
(
3
)
and based on the paraxial relationship, obtaining a focal length of the objective lens as:
f
=
Y
❘
"\[LeftBracketingBar]"
tan
(
Δ
θ
k
n
)
❘
"\[RightBracketingBar]"
+
❘
"\[LeftBracketingBar]"
tan
(
Δ
θ
k
1
)
❘
"\[RightBracketingBar]"
,
(
4
)
where Y is a length of the image plane,
Δ
θ
kn
=
θ
k
n
+
1
2
-
θ
k
n
is a view field angle at which the ray of wavenumber k n enters the objective lens, and
Δ
θ
k
1
=
θ
k
1
-
θ
k
n
+
1
2
is a view field angle at which the ray of wavenumber k 1 enters the objective lens; and
S 24 : defining a linearity evaluation coefficient R MS :
R
MS
=
∑
i
=
1
n
(
Δ
y
k
i
-
y
k
_
)
2
n
-
1
,
(
5
)
where Δy k i =y k i −y k i+1 is an interval between adjacent wavenumbers on the image plane, y k i =f·tan(Δθ k i ) is a y coordinate of the i-th wavenumber on the image plane, and y k =(y k i −y k i )/(n−1) is an average interval between equal-difference wavenumbers on the image plane.
9 . The design method of a wavenumber linearity dispersion optical system of claim 8 , wherein in the step S 4 , the compensation for distortion of the objective lens is
D
=
n
-
1
2
×
(
Δ
y
k
n
-
1
2
+
Δ
y
k
n
+
1
2
)
2
-
(
y
k
1
+
❘
"\[LeftBracketingBar]"
y
k
n
❘
"\[RightBracketingBar]"
)
2
(
y
k
1
+
❘
"\[LeftBracketingBar]"
y
k
n
❘
"\[RightBracketingBar]"
)
2
,
and
the compensation for the longitudinal chromatic aberration of the objective lens is C=y k i −|y k i |.
10 . An imaging spectrometer fabricated by the design method of a wavenumber linearity dispersion optical system of claim 1 .