Method for designing off-axial optical system with freeform surface

A method for designing three-dimensional freeform surface of an off-axial optical system comprising:

• step (S1), establishing a first initial system, wherein the first initial system comprises a plurality of initial surfaces, and each of the plurality of initial surfaces corresponds to one freeform surface of the off-axial optical system;
• step (S2), a Nth freeform surface of the off-axial optical system that need to be solved is selected, wherein N=1, keeping the plurality of initial surfaces unchanged, selecting a plurality of feature rays Ri (i=1, 2 . . . K), calculating a Nth plurality of intersections of the plurality of feature rays Ri (i=1, 2 . . . K) with the Nth freeform surface point by point based on a given object-image relationship and a vector form of Snell's law, wherein the Nth plurality of intersections are a Nth plurality of feature data points Pi (i=1, 2 . . . K) of the Nth freeform surface;
• and obtaining a Nth equation of the Nth freeform surface by surface fitting the Nth plurality of feature data points Pi (i=1, 2 . . . K), wherein the Nth equation of the Nth freeform surface comprises a Nth base conic term and a Nth freeform surface term;
• wherein the surface fitting the Nth plurality of feature data points Pi (i=1, 2 . . . K) comprises:
• step (21):
• surface fitting the Nth plurality of feature data points Pi (i=1, 2 . . . K) to a sphere in a first three-dimensional rectangular coordinates system, and obtaining a curvature c of the sphere and a center of curvature (xc, yc, zc) corresponding to the curvature c of the sphere;
• step (S22):
• defining a feature data point (xo, yo, zo) corresponding to a chief ray of a central field angle among entire field-of-view as a vertex of the sphere, defining a second three-dimensional rectangular coordinates system by the vertex of the sphere as origin and a line passing through the center of curvature and the vertex of the sphere as a Z′-axis;
• step (S23):
• transforming a first coordinates (xi, yi, zi) and a first normal vectors (αi, βi, γi), of the Nth plurality of feature data points Pi (i=1, 2 . . . K) in the first three-dimensional rectangular coordinates system, into a second coordinates (x′i, y′i, z′i) and a second normal vectors (α′i, β′i, γ′i), of the Nth plurality of feature data points Pi (i=1, 2 . . . K) in the second three-dimensional rectangular coordinates system;
• wherein a relationship between the second coordinates (x′i, y′i, z′i) and the first coordinates (xi, yi, zi) satisfies the following equation:
• {
• x
• i
• ′
• =
• x
• i
• -
• x
• o
• y
• i
• ′
• =
• (
• y
• i
• -
• y
• o
• )
• ⁢
• cos
• ⁢
• ⁢
• θ
• -
• (
• z
• i
• -
• z
• o
• )
• ⁢
• sin
• ⁢
• ⁢
• θ
• z
• i
• ′
• =
• (
• y
• i
• -
• y
• o
• )
• ⁢
• sin
• ⁢
• ⁢
• θ
• +
• (
• z
• i
• -
• z
• o
• )
• ⁢
• cos
• ⁢
• ⁢
• θ
• ;
• wherein θ is a tilt angle of the sphere, in a Y′O′Z′ plane of the second three-dimensional rectangular coordinates system relative to in a YOZ plane of the first three-dimensional rectangular coordinates system;
• step (S24):
• fitting the Nth plurality of feature data points Pi (i=1, 2 . . . K) into a conic surface equation of a conic surface in the second three-dimensional rectangular coordinates system, based on the second coordinates (x′i, y′i, z′i) and the curvature c of the sphere, and obtaining a conic constant k;
• and
• step (S25):
• removing a plurality of third coordinates and a plurality of third normal vectors of the Nth plurality of feature data points Pi (i=1, 2 . . . K), on the conic surface in the second three-dimensional rectangular coordinates system, from the second coordinates (x′i, y′i, z′i) and the second normal vectors (a′i, β′i, γ′i), to obtain a plurality of residual coordinates (x″i, y″i, z″i) (i=1, 2, . . . , K) and a plurality of residual normal vectors Ni=(a″i, β″i, −1) (i=1, 2, . . . , K);
• and surface fitting the plurality of residual coordinates (x″i, y″i, z″i) (i=1, 2, . . . , K) and the plurality of residual normal vectors Ni=(α″i, β″i, −1) (i=1, 2, . . . , K) to obtain a polynomial surface equation;
• the Nth equation of the Nth freeform surface are obtained by adding the conic surface equation and the polynomial surface equation;
• step (S3), keeping all the Nth freeform surface obtained in step (S2) and some of the plurality of initial surfaces that does not corresponds to the Nth freeform surface unchanged, and selecting N as N=N+1;
• step (S4), repeating the step (S2) to step (S3) until all freeform surfaces of the off-axial optical system that need to be solved are obtained, and obtaining a before-iteration off-axial optical system;
• and
• step (S5), taking the before-iteration off-axial optical system obtained in step (S4) as a second initial system for an iteration process, and obtaining an after-iteration off-axial optical system with improved imaging quality by multiple iterations.

A method for designing three-dimensional freeform surface of an off-axial optical system comprising:

• step (S1), establishing a first initial system, wherein the first initial system comprises a plurality of initial surfaces, and each of the plurality of initial surfaces corresponds to one freeform surface of the off-axial optical system;
• step (S2), a Nth freeform surface of the off-axial optical system that need to be solved is selected, wherein N=1, keeping the plurality of initial surfaces unchanged, selecting a plurality of feature rays Ri (i=1, 2 . . . K), calculating a Nth plurality of intersections of the plurality of feature rays Ri (i=1, 2 . . . K) with the Nth freeform surface point by point based on a given object-image relationship and a vector form of Snell's law, wherein the Nth plurality of intersections are a Nth plurality of feature data points Pi (i=1, 2 . . . K) of the Nth freeform surface;
• and obtaining a Nth equation of the Nth freeform surface by surface fitting the Nth plurality of feature data points Pi (i=1, 2 . . . K), wherein the Nth equation of the Nth freeform surface comprises a Nth base conic term and a Nth freeform surface term;
• wherein the surface fitting the Nth plurality of feature data points Pi (i=1, 2 . . . K) comprises:
• step (21):
• surface fitting the Nth plurality of feature data points Pi (i=1, 2 . . . K) to a sphere in a first three-dimensional rectangular coordinates system, and obtaining a curvature c of the sphere and a center of curvature (xc, yc, zc) corresponding to the curvature c of the sphere;
• step (S22):
• defining a feature data point (xo, yo, zo) corresponding to a chief ray of a central field angle among entire field-of-view as a vertex of the sphere, defining a second three-dimensional rectangular coordinates system by the vertex of the sphere as origin and a line passing through the center of curvature and the vertex of the sphere as a Z′-axis;
• step (S23):
• transforming a first coordinates (xi, yi, zi) and a first normal vectors αi, βi, γi), of the Nth plurality of feature data points Pi (i=1, 2 . . . K) in the first three-dimensional rectangular coordinates system, into a second coordinates (x′i, y′i, z′i) and a second normal vectors (α′i, β′i, γ′i), of the Nth plurality of feature data points Pi (i=1, 2 . . . K) in the second three-dimensional rectangular coordinates system;
• wherein a relationship between the second normal vectors (α′i, β′i, γ′i) and the first normal vectors (αi, βi, γi) satisfies the following equation:
• {
• α
• i
• ′
• =
• α
• i
• β
• i
• ′
• =
• β
• i
• ⁢
• cos
• ⁢
• ⁢
• θ
• -
• γ
• i
• ⁢
• sin
• ⁢
• ⁢
• θ
• γ
• i
• ′
• =
• β
• i
• ⁢
• sin
• ⁢
• ⁢
• θ
• +
• γ
• i
• ⁢
• cos
• ⁢
• ⁢
• θ
• ,
• wherein θ is a tilt angle of the sphere, in a Y′O′Z′ plane of the second three-dimensional rectangular coordinates system relative to in a YOZ plane of the first three-dimensional rectangular coordinates system;
• step (S24):
• fitting the Nth plurality of feature data points Pi (i=1, 2 . . . K) into a conic surface equation of a conic surface in the second three-dimensional rectangular coordinates system, based on the second coordinates (x′i, y′i, z′i) and the curvature c of the sphere, and obtaining a conic constant k;
• and
• step (S25):
• removing a plurality of third coordinates and a plurality of third normal vectors of the Nth plurality of feature data points Pi (i=1, 2 . . . K), on the conic surface in the second three-dimensional rectangular coordinates system, from the second coordinates (x′i, y′i, z′i) and the second normal vectors (α′i, β′i, γ′i), to obtain a plurality of residual coordinates (x″i, y″i, z″i) (i=1, 2, . . . , K) and a plurality of residual normal vectors Ni=(α″i, β″i, −1) (i=1, 2, . . . , K);
• and surface fitting the plurality of residual coordinates (x″i, y″i, z″i) (i=1, 2, . . . , K) and the plurality of residual normal vectors Ni=(a″i, β″i, −1) (i=1, 2, . . . , K) to obtain a polynomial surface equation;
• the Nth equation of the Nth freeform surface are obtained by adding the conic surface equation and the polynomial surface equation;
• step (S3), keeping all the Nth freeform surface obtained in step (S2) and some of the plurality of initial surfaces that does not corresponds to the Nth freeform surface unchanged, and selecting N as N=N+1;
• step (S4), repeating the step (S2) to step (S3) until all freeform surfaces of the off-axial optical system that need to be solved are obtained, and obtaining a before-iteration off-axial optical system;
• and
• step (S5), taking the before-iteration off-axial optical system obtained in step (S4) as a second initial system for an iteration process, and obtaining an after-iteration off-axial optical system with improved imaging quality by multiple iterations.