Method for designing freeform surfaces off-axial imaging system with a real exit pupil

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 first initial surfaces;
• step (S2), a surface in front of and adjacent to the real exit pupil is defined as a surface M, selecting a plurality of feature rays Ri (i=1, 2 . . . K), and calculating a plurality of intersections of the plurality of feature rays Ri (i=1, 2 . . . K) with the surface M;
• step (S3), calculating a Nth plurality of intersections of the plurality of feature rays Ri (i=1, 2 . . . K) with a before-iteration Nth freeform surface in front of the surface M point by point based on a given light mapping relationship and a vector form of Snell's law, wherein N=1, and the Nth plurality of intersections are a Nth plurality of feature data points Pi (i=1, 2 . . . K) of the before-iteration Nth freeform surface in front of the surface M;
• obtaining a Nth equation of the before-iteration Nth freeform surface by surface fitting the Nth plurality of feature data points Pi (i=1, 2 . . . K), wherein the Nth equation of the before-iteration 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 (31):
• 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 (S32):
• 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 (S33):
• 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 (S34):
• fitting the Nth plurality of feature data points Pi (i=l, 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 (S35):
• 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=(α″i, β″i, −1) (i=1, 2, . . . , K) to obtain a polynomial surface equation;
• the Nth equation of the before-iteration Nth freeform surface are obtained by adding the conic surface equation and the polynomial surface equation;
• step (S4), selecting N as N=N+1, and obtaining all before-iteration freeform surfaces by repeating step (S3) to step (S4);
• step (S5), taking the before-iteration freeform surfaces as a plurality of second initial surfaces, and obtaining all after-iteration freeform surfaces with improved imaging quality in front of the surface M by multiple iterations;
• step (S6), calculating a plurality of first intersections of the plurality of feature rays Ri (i=1, 2 . . . K) with the surface M point by point based on a given object-image relationship and the vector form of Snell's law, wherein the plurality of first intersections are a plurality of first feature data points of the surface M;
• obtaining a before-iteration surface M by surface fitting the plurality of first feature data points;
• and taking the before-iteration surface M as a third initial surface, and obtaining an after-iteration surface M by multiple iterations;
• and
• step (S7), taking the after-iteration surface M obtained in step (S6) as a fourth initial surface, keeping some of the plurality of first initial surfaces that does not corresponds to the after-iteration surface M unchanged;
• and repeating steps (S2)-(S6) until match two conditions:
• condition (1), after being redirected by the after-iteration freeform surfaces in front of the surface M, a plurality of first actual intersections of the plurality of feature rays Ri (i=1, 2 . . . K) with the after-iteration surface M are substantially close to the plurality of intersections;
• and condition (2), after being redirected by all the after-iteration freeform surfaces in front of the surface M and the after-iteration surface M, a plurality of second actual intersections of the plurality of feature rays Ri (i=1, 2 . . . K) with an imaging surface are substantially close to a plurality of image points.

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 first initial surfaces;
• step (S2), a surface in front of and adjacent to the real exit pupil is defined as a surface M, selecting a plurality of feature rays Ri (i=1, 2 . . . K), and calculating a plurality of intersections of the plurality of feature rays Ri (i=1, 2 . . . K) with the surface M;
• step (S3), calculating a Nth plurality of intersections of the plurality of feature rays Ri (i=1, 2 . . . K) with a before-iteration Nth freeform surface in front of the surface M point by point based on a given light mapping relationship and a vector form of Snell's law, wherein N=1, and the Nth plurality of intersections are a Nth plurality of feature data points Pi (i=1, 2 . . . K) of the before-iteration Nth freeform surface in front of the surface M;
• obtaining a Nth equation of the before-iteration Nth freeform surface by surface fitting the Nth plurality of feature data points Pi (i=1, 2 . . . K), wherein the Nth equation of the before-iteration 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 (31):
• 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 (S32):
• 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 (S33):
• 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 (S34):
• 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 (S35):
• 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=(α″i, β″i, −1) (i=1, 2, . . . , K) to obtain a polynomial surface equation;
• the Nth equation of the before-iteration Nth freeform surface are obtained by adding the conic surface equation and the polynomial surface equation;
• step (S4), selecting N as N=N+1, and obtaining all before-iteration freeform surfaces by repeating step (S3) to step (S4);
• step (S5), taking the before-iteration freeform surfaces as a plurality of second initial surfaces, and obtaining all after-iteration freeform surfaces with improved imaging quality in front of the surface M by multiple iterations;
• step (S6), calculating a plurality of first intersections of the plurality of feature rays Ri (i=1, 2 . . . K) with the surface M point by point based on a given object-image relationship and the vector form of Snell's law, wherein the plurality of first intersections are a plurality of first feature data points of the surface M;
• obtaining a before-iteration surface M by surface fitting the plurality of first feature data points;
• and taking the before-iteration surface M as a third initial surface, and obtaining an after-iteration surface M by multiple iterations;
• and
• step (S7), taking the after-iteration surface M obtained in step (S6) as a fourth initial surface, keeping some of the plurality of first initial surfaces that does not corresponds to the after-iteration surface M unchanged;
• and repeating steps (S2)-(S6) until match two conditions:
• condition (1), after being redirected by the after-iteration freeform surfaces in front of the surface M, a plurality of first actual intersections of the plurality of feature rays Ri (i=1, 2 . . . K) with the after-iteration surface M are substantially close to the plurality of intersections;
• and condition (2), after being redirected by all the after-iteration freeform surfaces in front of the surface M and the after-iteration surface M, a plurality of second actual intersections of the plurality of feature rays Ri (i=1, 2 . . . K) with an imaging surface are substantially close to a plurality of image points.