# Patent

## Title
LED freeform surface illumination system

## Abstract

A LED freeform surface illumination system is provided. The LED freeform surface illumination system includes a LED point light source, a freeform surface lens, and a light receiving surface. A light emitting angle φ of the LED point light source is divided into three regions. The freeform surface lens includes a spherical surface and a freeform surface opposite to the spherical surface. The freeform surface comprises three annular regions. Each region of the three regions of the light emitting angle φ of the LED point light source corresponds with one annular region of the freeform surface. Lights of each region of the three regions of the light emitting angle φ of the LED point light source pass the corresponded annular region of the freeform surface to form one rectangular light spot.

## Background
CROSS-REFERENCE TO RELATED APPLICATIONS
This application claims priority to Chinese Patent Application No. 201410057436.4, filed on Feb. 20, 2014, the disclosure of which is incorporated herein by reference.
FIELD
The present disclosure relates to LED freeform surface illumination system.
BACKGROUND
It is difficult to achieve a uniform illumination on a surface in a required area by LED which is commonly seen as a point light source. It is necessary to redistribute the LED light by a secondary optical design for a LED illumination system. Conventionally, a lens assembly is used in the LED illumination system to achieve the uniform illumination.



BRIEF DESCRIPTION OF THE DRAWINGS
Implementations of the present technology will now be described, by way of example only, with reference to the attached figures.
FIG. 1 is flowchart of one embodiment of a design method of LED freeform surface illumination system.
FIG. 2 is a schematic view showing one embodiment of divided regions of light emitting angles of a LED point light source and annular divided regions of a freeform surface in the design method of LED freeform surface illumination system.
FIG. 3 is a schematic view showing one embodiment of light mapping from a point light source to a light receiving surface in the design method of LED freeform surface illumination system.
FIG. 4 is a schematic view showing one embodiment of a light redistribution design for LED point light source in the design method of LED freeform surface illumination system.
FIG. 5 is a three dimensional image of one embodiment of a freeform surface lens obtained by the design method of LED freeform surface illumination system.
FIG. 6 is flowchart of another embodiment of the design method of LED freeform surface illumination system.
FIG. 7 is a schematic view showing a structure and light paths of one embodiment of the LED freeform surface illumination system.
FIG. 8 is a LightTools light simulation image of one embodiment of the freeform surface lens with light emitting angles in a region of 0°-10°.
FIG. 9 is a LightTools light simulation image of the freeform surface lens with emitting angles in a region of 10°-20°.
FIG. 10 is a LightTools light simulation image of the freeform surface lens with emitting angles in a region of 20°-30°.
FIG. 11 is a diagram showing a software-simulated square light spot and an illuminance distribution thereof.
FIG. 12 is a diagram showing another software-simulated square light spot and illuminance distributions on X axis and Y axis thereof.



DETAILED DESCRIPTION
The disclosure is illustrated by way of example and not by way of limitation in the figures of the accompanying drawings in which like references indicate similar elements. It should be noted that references to “an” or “one” embodiment in this disclosure are not necessarily to the same embodiment, and such references mean “at least one”.
It will be appreciated that for simplicity and clarity of illustration, where appropriate, reference numerals have been repeated among the different figures to indicate corresponding or analogous elements. In addition, numerous specific details are set forth in order to provide a thorough understanding of the embodiments described herein. However, it will be understood by those of ordinary skill in the art that the embodiments described herein can be practiced without these specific details. In other instances, methods, procedures and components have not been described in detail so as not to obscure the related relevant feature being described. Also, the description is not to be considered as limiting the scope of the embodiments described herein. The drawings are not necessarily to scale and the proportions of certain parts may be exaggerated to better illustrate details and features of the present disclosure.
Several definitions that apply throughout this disclosure will now be presented.
The term “comprise” or “comprising” when utilized, means “include or including, but not necessarily limited to”; it specifically indicates open-ended inclusion or membership in the so-described combination, group, series, and the like.
Referring to FIG. 1, a flowchart is presented in accordance with an example embodiment as illustrated. The embodiment of a design method of LED freeform surface illumination system 100 is provided by way of example, as there are a variety of ways to carry out the method. The method 100 described below can be carried out using the configurations illustrated in FIGS. 2 to 5 for example, and various elements of these figures are referenced in explaining example method 100. Each block shown in FIG. 1 represents one or more processes, methods, or subroutines carried out in the exemplary method 100. Additionally, the illustrated order of blocks is by example only and the order of the blocks can be changed. The exemplary method 100 can begin at block 101. Depending on the embodiment, additional steps can be added, others removed, and the ordering of the steps can be changed.
At block 101, a light emitting angle φ (0≦φ≦φT) of a LED point light source 10 is divided into three regions. The three regions are 0≦φ≦φN, φN≦φ≦φS, and φS≦φ≦φT. Each region can respectively form a rectangular light spot on a light receiving surface 14. A length of the rectangular spot can be 2Lx. A width of the rectangular spot can be 2Ly. φT is the largest light emitting angle of the LED point light source 10.
At block 102, a mapping relationship of each region on the light receiving surface 14 is obtained.
At block 103, a light redistribution design for the LED point light source 10 is performed and a system of first order partial differential equations of a freeform surface 124 is achieved.
At block 104, a system of first order quasi linear differential equations of the freeform surface 124 is obtained according to the mapping relationships of the three regions on the light receiving surface 14 and the system of first order partial differential equations of the freeform surface 124. A plurality of freeform surface data is obtained by solving the system of first order quasi linear differential equations.
At block 105, the freeform surface 124 is obtained by surface fitting the plurality of freeform surface data.
Referring to FIG. 2, the largest light emitting angle φT of light emitted from the LED point light source 10 is not limited. In one embodiment, the largest light emitting angle φT is about 30°. The light emitting angle φ (0≦φ≦φT) of the light emitted from the LED point light source 10 can be divided into three regions such as three equal regions, three equal proportional regions, or three arbitrary regions. Specifically, the three regions are 0≦φ≦φN, φN≦φ≦φS, and φS≦φ≦φT. In one embodiment, the light emitting angle φ (0≦φ≦φT) of the light emitted from the LED point light source 10 is divided into three equal regions (e.g., φN=10°, φS=20°, and φT=30°).
In the region of the 0≦φ≦φN, the lights of φ=0° passing the freeform surface 124 reach the center of the rectangular light spot on the light receiving surface 14. The lights of φ=φN passing the freeform surface 124 reach the edges of the rectangular light spot on the light receiving surface 14. To keep the freeform surface 124 continuous, the lights at the junction between the region 0≦φ≦φN and the region φN≦φ≦φS cannot have a sudden change. Thus, the lights of φ=φN passing the freeform surface 124 reach the edges of the rectangular light spot on the light receiving surface 14, and the lights of φ=φS passing the freeform surface 124 reach the center of the rectangular light spot on the light receiving surface 14. Similarly, to keep the freeform surface 124 continuous, the lights at the junction between the region φN≦φ≦φS and the region φS≦φ≦φT cannot have a sudden change. Thus, the lights of φ=φS passing the freeform surface 124 reach the center of the rectangular light spot on the light receiving surface 14, and the lights of φ=φT passing the freeform surface 124 reach the edges of the rectangular light spot on the light receiving surface 14.
Referring to FIG. 3, the length 2Lx and the width 2Ly of the rectangular light spot are not limited. Lx and Ly are constant values that are predetermined. In one embodiment, the length 2Lx is equal to the width 2Ly that are both about 20 mm (i.e., Lx=Ly=10 mm).
The mapping relationship is established between the lights emitted from the LED point light source 10 having the light emitting angles in the three regions and the light spots on the light receiving surface 14. In the region of the 0≦φ≦φN, the lights emitted from the center of the LED point light source 10 (i.e., the lights of φ=0°) are mapped to the center of the rectangular light spot on the light receiving surface 14. The lights of φ=φN are mapped to the edges of the rectangular light spot on the light receiving surface 14. A half of the length of the rectangular light spot is Lx. A half of the width of the rectangular light spot is Ly. The Lx and the Ly can satisfy the following equation:




  
    
      
        
          
            π
            2
          
          ⁢
          
            
              ∫
              0
              
                φ
                N
              
            
            ⁢
            
              
                I
                0
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              φ
              ⁢
              
                
              
              ⁢
              sin
              ⁢
              
                
              
              ⁢
              φ
              ⁢
              
                
              
              ⁢
              
                d
                φ
              
            
          
        
        =
        
          E
          ×
          
            L
            x
          
          ×
          
            L
            y
          
        
      
    
    
      
        (
        
          2
          ⁢
          
            -
          
          ⁢
          1
        
        )
      
    
  




In the equation (2-1), E represents illuminance.

In the region 0≦φ≦φN, each light of the light emitting angle φ is mapped to a rectangular frame shaped light spot having a length 2X and a width 2Y on the light receiving surface 14. A half of the length of the rectangular frame shaped light spot is X. A half of the width of the rectangular frame shaped light spot is Y. According to the law of conservation of energy, the X and Y can satisfy the following equation:




  
    
      
        
          
            π
            2
          
          ⁢
          
            
              ∫
              0
              φ
            
            ⁢
            
              
                I
                0
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              φ
              ⁢
              
                
              
              ⁢
              sin
              ⁢
              
                
              
              ⁢
              φ
              ⁢
              
                
              
              ⁢
              
                d
                φ
              
            
          
        
        =
        
          E
          ×
          X
          ×
          Y
        
      
    
    
      
        (
        
          2
          ⁢
          
            -
          
          ⁢
          2
        
        )
      
    
  




According to the equation (2-1) and the equation (2-2), the mapping relationship in the region 0≦φ≦φN of the light emitting angle φ can be expressed as the following system of equations:




  
    
      
        {
        
          
            
              
                X
                =
                
                  
                    L
                    x
                  
                  ⁢
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      φ
                    
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        φ
                        N
                      
                    
                  
                
              
            
          
          
            
              
                Y
                =
                
                  
                    L
                    y
                  
                  ⁢
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      φ
                    
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      
                        φ
                        N
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        
          2
          ⁢
          
            -
          
          ⁢
          3
        
        )
      
    
  




Similarly, the mapping relationship in the region φN≦φ≦φS of the light emitting angle φ can be expressed as the following system of equations:




  
    
      
        {
        
          
            
              
                X
                =
                
                  
                    L
                    x
                  
                  ⁢
                  
                    
                      
                        
                          
                            sin
                            2
                          
                          ⁢
                          
                            φ
                            S
                          
                        
                        -
                        
                          
                            sin
                            2
                          
                          ⁢
                          φ
                        
                      
                      
                        
                          
                            sin
                            2
                          
                          ⁢
                          
                            φ
                            S
                          
                        
                        -
                        
                          
                            sin
                            2
                          
                          ⁢
                          
                            φ
                            N
                          
                        
                      
                    
                  
                
              
            
          
          
            
              
                Y
                =
                
                  
                    L
                    y
                  
                  ⁢
                  
                    
                      
                        
                          
                            sin
                            2
                          
                          ⁢
                          
                            φ
                            S
                          
                        
                        -
                        
                          
                            sin
                            2
                          
                          ⁢
                          φ
                        
                      
                      
                        
                          
                            sin
                            2
                          
                          ⁢
                          
                            φ
                            S
                          
                        
                        -
                        
                          
                            sin
                            2
                          
                          ⁢
                          
                            φ
                            N
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        
          2
          ⁢
          
            -
          
          ⁢
          4
        
        )
      
    
  




Similarly, the mapping relationship in the region φS≦φ≦φT of the light emitting angle φ can be expressed as the following system of equations:




  
    
      
        {
        
          
            
              
                X
                =
                
                  
                    L
                    x
                  
                  ⁢
                  
                    
                      
                        
                          
                            sin
                            2
                          
                          ⁢
                          φ
                        
                        -
                        
                          
                            sin
                            2
                          
                          ⁢
                          
                            φ
                            S
                          
                        
                      
                      
                        
                          
                            sin
                            2
                          
                          ⁢
                          
                            φ
                            T
                          
                        
                        -
                        
                          
                            sin
                            2
                          
                          ⁢
                          
                            φ
                            S
                          
                        
                      
                    
                  
                
              
            
          
          
            
              
                Y
                =
                
                  
                    L
                    y
                  
                  ⁢
                  
                    
                      
                        
                          
                            sin
                            2
                          
                          ⁢
                          φ
                        
                        -
                        
                          
                            sin
                            2
                          
                          ⁢
                          
                            φ
                            S
                          
                        
                      
                      
                        
                          
                            sin
                            2
                          
                          ⁢
                          
                            φ
                            T
                          
                        
                        -
                        
                          
                            sin
                            2
                          
                          ⁢
                          
                            φ
                            S
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        
          2
          ⁢
          
            -
          
          ⁢
          5
        
        )
      
    
  




According to a equal proportion relationship, a relationship between the any point (x, y) on the light receiving surface 14 and the length and width (2X and 2Y) of the rectangular frame shaped light spot can be expressed as the following system of equations:




  
    
      
        {
        
          
            
              
                0
                ≤
                θ
                ≤
                
                  π
                  /
                  4
                
              
            
            
              
                {
                
                  
                    
                      
                        x
                        =
                        X
                      
                    
                  
                  
                    
                      
                        y
                        =
                        
                          
                            
                              4
                              ⁢
                              θ
                            
                            π
                          
                          ⁢
                          Y
                        
                      
                    
                  
                
              
            
          
          
            
              
                
                  π
                  /
                  4
                
                ≤
                θ
                ≤
                
                  π
                  /
                  2
                
              
            
            
              
                {
                
                  
                    
                      
                        x
                        =
                        
                          
                            (
                            
                              2
                              -
                              
                                
                                  4
                                  ⁢
                                  θ
                                
                                π
                              
                            
                            )
                          
                          ⁢
                          X
                        
                      
                    
                  
                  
                    
                      
                        y
                        =
                        Y
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        
          2
          ⁢
          
            -
          
          ⁢
          6
        
        )
      
    
  




According to the equations (2-1) to (2-6), the mapping relationship in the region 0≦φ≦φT of the light emitting angle φ can be expressed as the following system of equations:




  
    
      
        {
        
          
            
              
                X
                ,
                
                  Y
                  =
                  
                    {
                    
                      
                        
                          
                            
                              L
                              x
                            
                            ,
                            
                              
                                L
                                y
                              
                              ⁢
                              
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  φ
                                
                                
                                  sin
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    φ
                                    N
                                  
                                
                              
                            
                          
                        
                        
                          
                            0
                            ≤
                            φ
                            ≤
                            
                              φ
                              N
                            
                          
                        
                      
                      
                        
                          
                            
                              L
                              x
                            
                            ,
                            
                              
                                L
                                y
                              
                              ⁢
                              
                                
                                  
                                    
                                      
                                        sin
                                        2
                                      
                                      ⁢
                                      
                                        φ
                                        S
                                      
                                    
                                    -
                                    
                                      
                                        sin
                                        2
                                      
                                      ⁢
                                      φ
                                    
                                  
                                  
                                    
                                      
                                        sin
                                        2
                                      
                                      ⁢
                                      
                                        φ
                                        S
                                      
                                    
                                    -
                                    
                                      
                                        sin
                                        2
                                      
                                      ⁢
                                      
                                        φ
                                        N
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        
                          
                            
                              φ
                              N
                            
                            ≤
                            φ
                            ≤
                            
                              φ
                              S
                            
                          
                        
                      
                      
                        
                          
                            
                              L
                              x
                            
                            ,
                            
                              
                                L
                                y
                              
                              ⁢
                              
                                
                                  
                                    
                                      
                                        sin
                                        2
                                      
                                      ⁢
                                      φ
                                    
                                    -
                                    
                                      
                                        sin
                                        2
                                      
                                      ⁢
                                      
                                        φ
                                        S
                                      
                                    
                                  
                                  
                                    
                                      
                                        sin
                                        2
                                      
                                      ⁢
                                      
                                        φ
                                        T
                                      
                                    
                                    -
                                    
                                      
                                        sin
                                        2
                                      
                                      ⁢
                                      
                                        φ
                                        S
                                      
                                    
                                  
                                
                              
                            
                          
                        
                        
                          
                            
                              φ
                              S
                            
                            ≤
                            φ
                            ≤
                            
                              φ
                              T
                            
                          
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    
                      
                    
                  
                
                
                  
                    
                      
                    
                  
                
                
                  
                    
                      
                    
                  
                
              
            
          
          
            
              
                {
                
                  
                    
                      
                        
                          
                            x
                            =
                            X
                          
                        
                      
                      
                        
                          
                            y
                            =
                            
                              
                                
                                  4
                                  ⁢
                                  θ
                                
                                π
                              
                              ⁢
                              Y
                            
                          
                        
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    0
                  
                  ≤
                  θ
                  ≤
                  
                    π
                    /
                    4
                  
                
              
            
            
              
                
              
            
          
          
            
              
                {
                
                  
                    
                      
                        
                          
                            x
                            =
                            
                              
                                (
                                
                                  2
                                  -
                                  
                                    
                                      4
                                      ⁢
                                      θ
                                    
                                    π
                                  
                                
                                )
                              
                              ⁢
                              X
                            
                          
                        
                      
                      
                        
                          
                            y
                            =
                            Y
                          
                        
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      π
                      /
                      4
                    
                  
                  ≤
                  θ
                  ≤
                  
                    π
                    /
                    2
                  
                
              
            
            
              
                
              
            
          
        
      
    
    
      
        (
        
          2
          ⁢
          
            -
          
          ⁢
          7
        
        )
      
    
  




At block 103, to achieve a uniform rectangular light spot on the light receiving surface 14, the light redistribution design is performed for redistributing the lights emitted from the LED point light source 10. Referring to FIG. 4, a spherical coordinate system can be defined as (θ, φ, ρ). In the spherical coordinate system, ρ is the radial distance between a point and the LED point light source 10, θ is the azimuthal angle, and the light emitting angle φ is the polar angle. I represents incident light. A vector I can be expressed as the following equation:

I=(sin φ cos θ,sin φ sin θ,cos φ)  (3-1)

P represents any point on the freeform surface 124. A vector P can be expressed as the following equation:

P=ρ×I=(ρ sin φ cos θ,ρ sin φ sin θ,ρ cos φ)  (3-2))

O represents an emitting light. A vector O can be expressed as the following equation:

O=(T−P)/|T−P|  (3-3)

In the equation (3-3), T=(x,y,z)
A normal vector N can be expressed as the following equation:

N=PθPφ/|PθPφ|=(Nx,Ny,Nz)  (3-4)

According to the equations (3-1) to (3-4), the normal vector N can be expressed as the following system of equations:




  
    
      
        {
        
          
            
              
                
                  N
                  x
                
                =
                
                  
                    
                      
                        -
                        
                          ρ
                          θ
                        
                      
                      ×
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                    +
                    
                      
                        ρ
                        φ
                      
                      ×
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      φ
                      ×
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      φ
                      ×
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                    -
                    
                      ρ
                      ×
                      
                        sin
                        2
                      
                      ⁢
                      φ
                      ×
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                  
                  
                    
                      (
                      
                        
                          ρ
                          θ
                          2
                        
                        +
                        
                          
                            ρ
                            φ
                            2
                          
                          ⁢
                          
                            sin
                            2
                          
                          ⁢
                          φ
                        
                        +
                        
                          
                            ρ
                            2
                          
                          ⁢
                          
                            sin
                            2
                          
                          ⁢
                          φ
                        
                      
                      )
                    
                    
                      1
                      /
                      2
                    
                  
                
              
            
          
          
            
              
                
                  N
                  y
                
                =
                
                  
                    
                      
                        ρ
                        θ
                      
                      ×
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                    +
                    
                      
                        ρ
                        φ
                      
                      ×
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      φ
                      ×
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      φ
                      ×
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                    -
                    
                      ρ
                      ×
                      
                        sin
                        2
                      
                      ⁢
                      φ
                      ×
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                    
                  
                  
                    
                      (
                      
                        
                          ρ
                          θ
                          2
                        
                        +
                        
                          
                            ρ
                            φ
                            2
                          
                          ⁢
                          
                            sin
                            2
                          
                          ⁢
                          φ
                        
                        +
                        
                          
                            ρ
                            2
                          
                          ⁢
                          
                            sin
                            2
                          
                          ⁢
                          φ
                        
                      
                      )
                    
                    
                      1
                      /
                      2
                    
                  
                
              
            
          
          
            
              
                
                  N
                  z
                
                =
                
                  
                    
                      
                        -
                        
                          ρ
                          θ
                        
                      
                      ×
                      
                        sin
                        2
                      
                      ⁢
                      φ
                    
                    -
                    
                      ρ
                      ×
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      φ
                      ×
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      φ
                    
                  
                  
                    
                      (
                      
                        
                          ρ
                          θ
                          2
                        
                        +
                        
                          
                            ρ
                            φ
                            2
                          
                          ⁢
                          
                            sin
                            2
                          
                          ⁢
                          φ
                        
                        +
                        
                          
                            ρ
                            2
                          
                          ⁢
                          
                            sin
                            2
                          
                          ⁢
                          φ
                        
                      
                      )
                    
                    
                      1
                      /
                      2
                    
                  
                
              
            
          
        
      
    
    
      
        (
        
          3
          ⁢
          
            -
          
          ⁢
          5
        
        )
      
    
  




The vector I, the vector O and the vector N follow the Snell's law, that is satisfying the following equation:

no×O−nI×I=[no2+nI2−2×no×nI×(O·I)]1/2×N  (3-6)

In the equation (3-6), the no represents a refractive index of air, the nI represents a refractive index of a freeform surface lens 12.

The system of first order partial differential equations of the freeform surface 124 can be expressed as:




   
  
    
      
        
          {
          
            
              
                
                  
                    ρ
                    φ
                  
                  =
                  
                    
                      ρ
                      ⁡
                      
                        (
                        
                          θ
                          ,
                          φ
                        
                        )
                      
                    
                    ×
                    
                      
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                          ×
                          
                            (
                            
                              
                                
                                  n
                                  0
                                
                                ×
                                
                                  O
                                  z
                                
                              
                              -
                              
                                
                                  n
                                  I
                                
                                ×
                                
                                  I
                                  z
                                
                              
                            
                            )
                          
                        
                        -
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                          ×
                          A
                        
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                          ×
                          
                            (
                            
                              
                                
                                  n
                                  0
                                
                                ×
                                
                                  O
                                  z
                                
                              
                              -
                              
                                
                                  n
                                  I
                                
                                ×
                                
                                  I
                                  z
                                
                              
                            
                            )
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                          ×
                          A
                        
                      
                    
                  
                
              
            
            
              
                
                  
                    ρ
                    θ
                  
                  =
                  
                    
                      ρ
                      ⁡
                      
                        (
                        
                          θ
                          ,
                          φ
                        
                        )
                      
                    
                    ×
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                    ×
                    
                      
                        
                          
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                                ×
                                
                                  (
                                  
                                    
                                      
                                        n
                                        0
                                      
                                      ×
                                      
                                        O
                                        x
                                      
                                    
                                    -
                                    
                                      
                                        n
                                        I
                                      
                                      ×
                                      
                                        I
                                        x
                                      
                                    
                                  
                                  )
                                
                              
                              -
                            
                          
                        
                        
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                              ×
                              
                                (
                                
                                  
                                    
                                      n
                                      0
                                    
                                    ×
                                    
                                      O
                                      y
                                    
                                  
                                  -
                                  
                                    
                                      n
                                      I
                                    
                                    ×
                                    
                                      I
                                      y
                                    
                                  
                                
                                )
                              
                            
                          
                        
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                          ×
                          
                            (
                            
                              
                                
                                  n
                                  0
                                
                                ×
                                
                                  O
                                  z
                                
                              
                              -
                              
                                
                                  n
                                  I
                                
                                ×
                                
                                  I
                                  z
                                
                              
                            
                            )
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                          ×
                          A
                        
                      
                    
                  
                
              
            
            
              
                
                  A
                  =
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                      ×
                      
                        (
                        
                          
                            
                              n
                              0
                            
                            ×
                            
                              O
                              y
                            
                          
                          -
                          
                            
                              n
                              I
                            
                            ×
                            
                              I
                              y
                            
                          
                        
                        )
                      
                    
                    +
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      θ
                      ×
                      
                        (
                        
                          
                            
                              n
                              0
                            
                            ×
                            
                              O
                              x
                            
                          
                          -
                          
                            
                              n
                              I
                            
                            ×
                            
                              I
                              x
                            
                          
                        
                        )
                      
                    
                  
                
              
            
          
        
      
      
        
          (
          
            3
            ⁢
            
              -
            
            ⁢
            7
          
          )
        
      
    
  




At block 104, the system of first order quasi-linear partial differential equations of the freeform surface 124 can be obtained by substituting the equation (2-7) into the equation (3-7). The system of first order quasi-linear partial differential equations can be simplified and solved by using the Runge-Kutta method and Lax difference scheme. A plurality of freeform surface data of the freeform surface 124 can be obtained by using MATLAB programming according to the system of first order quasi-linear partial differential equations.
At block 105, the freeform surface 124 can be obtained by surface fitting the plurality of freeform surface data. The surface fitting can be realized by using non uniform rational B-spline (NURBS). Referring to FIG. 5, in one embodiment, the surface fitting can be realized by using a rhinoceros modeling software based on a principle of the NURBS to obtain a lens comprising the freeform surface 124.
In the design method of LED freeform surface illumination system 100, the light emitting angle φ of the LED point light source 10 is not limited to be divided into three regions. The light emitting angle φ of the LED point light source 10 can be divided into at least two regions. Referring to FIG. 6, a flowchart is presented in accordance with an example embodiment as illustrated. The embodiment of the design method of LED freeform surface illumination system 200 is provided
At block 201, a light emitting angle φ (0≦φ≦φT) of a LED point light source 10 is divided into two regions. Each region can respectively form a rectangular light spot on a light receiving surface 14. A length of the rectangular spot can be 2Lx. A width of the rectangular spot can be 2Ly. A light emitting angle φT is the largest light emitting angle of the LED point light source 10.
At block 202, a mapping relationship of each region on the light receiving surface 14 is obtained.
At block 203, a light redistribution design for the LED point light source 10 is performed and the system of first order partial differential equations of a freeform surface 124 is achieved.
At block 204, a system of first order quasi linear differential equations of the freeform surface 124 is obtained according to the mapping relationship of the two regions in the light receiving surface 14 and the system of first order partial differential equations of the freeform surface 124. A plurality of freeform surface data is obtained by solving the system of first order quasi linear differential equations.
At block 205, the freeform surface 124 is obtained by surface fitting the plurality of freeform surface data.
Referring to FIG. 7, in one embodiment, a LED freeform surface illumination system 300 based on the above described design method is provided. The LED freeform surface illumination system 300 comprises a LED point light source 10, a freeform surface lens 12, and a light receiving surface 14.
The space the LED freeform surface illumination system 300 located in, is defined by a three-dimensional Cartesian coordinate system (x, y, z) and a spherical coordinate system (θ, φ, ρ). The origin of the three-dimensional Cartesian coordinate system (x, y, z) and the origin of the spherical coordinate system (θ, φ, ρ) are at the same point.
The LED point light source 10 is located at the origin (0, 0, 0) of the three-dimensional Cartesian coordinate system (x, y, z). The LED point light source 10 has a largest light emitting angle φT. In one embodiment, the largest light emitting angle φT is 30°. The lights having the light emitting angle φ of 0° of the LED point light source 10 is along the z-axis of the three-dimensional Cartesian coordinate system (x, y, z). A diameter of the LED point light source 10 can be about 1 mm to about 5 mm. The light emitting angle φ (0≦φ≦φT) of the light emitted from the LED point light source 10 can be divided into three regions such as three equal regions, three equal proportional regions, or three arbitrary regions. Specifically, the three regions are 0≦φ≦φN, φN≦φ≦φS, and φS≦φ≦φT. In one embodiment, the light emitting angle φ (0≦φ≦φT) of the light emitted from the LED point light source 10 is divided into three equal regions, (e.g., φN=10°, φS=20°, and φT=30°).
The axis of the freeform surface lens 12 is along the z axis of the three-dimensional Cartesian coordinate system (x, y, z). The freeform surface lens 12 comprises a spherical surface 122 and the freeform surface 124 opposite to the spherical surface 122. The spherical surface 122 is a light incident surface. The freeform surface 124 is a light emitting surface.
A spherical centre of the spherical surface 122 is at the origin (0, 0, 0). Thus, the incident lights I will not be refracted when passing the spherical surface 122. The radius of the spherical surface 122 is not limited.
The freeform surface 124 comprises three annular regions. Each region of the light emitting angle φ of the LED point light source 10 corresponds with one annular region of the freeform surface 124. The lights of each region of the light emitting angle φ of the LED point light source 10 can pass the corresponded annular region of the freeform surface 124, to form one rectangular light spot on the light receiving surface 14. The length of the rectangular light spot can be 2Lx. The width of the rectangular light spot can be 2Ly. The system of first order partial differential equations of the freeform surface 124 in the spherical coordinate system (θ, φ, ρ) can be expressed as the following equations:




  {
  
    
      
        
          
            
              
                ρ
                φ
              
              =
              
                
                  ρ
                  ⁡
                  
                    (
                    
                      θ
                      ,
                      φ
                    
                    )
                  
                
                ×
                
                  
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      φ
                      ×
                      
                        (
                        
                          
                            
                              n
                              0
                            
                            ×
                            
                              O
                              z
                            
                          
                          -
                          
                            
                              n
                              I
                            
                            ×
                            
                              I
                              z
                            
                          
                        
                        )
                      
                    
                    -
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      φ
                      ×
                      A
                    
                  
                  
                    
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      φ
                      ×
                      
                        (
                        
                          
                            
                              n
                              0
                            
                            ×
                            
                              O
                              z
                            
                          
                          -
                          
                            
                              n
                              I
                            
                            ×
                            
                              I
                              z
                            
                          
                        
                        )
                      
                    
                    +
                    
                      sin
                      ⁢
                      
                        
                      
                      ⁢
                      φ
                      ×
                      A
                    
                  
                
              
            
          
        
        
          
            
              
                
                  ρ
                  θ
                
                =
                
                  
                    ρ
                    ⁡
                    
                      (
                      
                        θ
                        ,
                        φ
                      
                      )
                    
                  
                  ×
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  φ
                  ×
                  
                    
                      
                        
                          
                            
                              sin
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                              ×
                              
                                (
                                
                                  
                                    
                                      n
                                      0
                                    
                                    ×
                                    
                                      O
                                      x
                                    
                                  
                                  -
                                  
                                    
                                      n
                                      I
                                    
                                    ×
                                    
                                      I
                                      x
                                    
                                  
                                
                                )
                              
                            
                            -
                          
                        
                      
                      
                        
                          
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            θ
                            ×
                            
                              (
                              
                                
                                  
                                    n
                                    0
                                  
                                  ×
                                  
                                    O
                                    y
                                  
                                
                                -
                                
                                  
                                    n
                                    I
                                  
                                  ×
                                  
                                    I
                                    y
                                  
                                
                              
                              )
                            
                          
                        
                      
                    
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                        ×
                        
                          (
                          
                            
                              
                                n
                                0
                              
                              ×
                              
                                O
                                z
                              
                            
                            -
                            
                              
                                n
                                I
                              
                              ×
                              
                                I
                                z
                              
                            
                          
                          )
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                        ×
                        A
                      
                    
                  
                
              
              ;
            
          
        
      
      ⁢
      
        

      
      ⁢
      wherein
      ⁢
      
        

      
      ⁢
      A
    
    =
    
      
        
          sin
          ⁢
          
            
          
          ⁢
          θ
          ×
          
            (
            
              
                
                  n
                  0
                
                ×
                
                  O
                  y
                
              
              -
              
                
                  n
                  I
                
                ×
                
                  I
                  y
                
              
            
            )
          
        
        +
        
          cos
          ⁢
          
            
          
          ⁢
          θ
          ×
          
            (
            
              
                
                  n
                  0
                
                ×
                
                  O
                  x
                
              
              -
              
                
                  n
                  I
                
                ×
                
                  I
                  x
                
              
            
            )
          
          ⁢
          
            

          
          ⁢
          
            O
            x
          
        
      
      =
      
        
          
            
              x
              -
              
                ρ
                ⁢
                
                  
                
                ⁢
                sin
                ⁢
                
                  
                
                ⁢
                φ
                ⁢
                
                  
                
                ⁢
                cos
                ⁢
                
                  
                
                ⁢
                θ
              
            
            
              
                
                  
                    (
                    
                      x
                      -
                      
                        ρ
                        ⁢
                        
                          
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                        ⁢
                        
                          
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                    
                    )
                  
                  2
                
                +
                
                  
                    (
                    
                      y
                      -
                      
                        ρ
                        ⁢
                        
                          
                        
                        ⁢
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                        ⁢
                        
                          
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        θ
                      
                    
                    )
                  
                  2
                
                +
                
                  
                    (
                    
                      
                        z
                        o
                      
                      -
                      
                        ρ
                        ⁢
                        
                          
                        
                        ⁢
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                      
                    
                    )
                  
                  2
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            O
            y
          
        
        =
        
          
            
              
                y
                -
                
                  ρ
                  ⁢
                  
                    
                  
                  ⁢
                  sin
                  ⁢
                  
                    
                  
                  ⁢
                  φ
                  ⁢
                  
                    
                  
                  ⁢
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  θ
                
              
              
                
                  
                    
                      (
                      
                        x
                        -
                        
                          ρ
                          ⁢
                          
                            
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                          ⁢
                          
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          θ
                        
                      
                      )
                    
                    2
                  
                  +
                  
                    
                      (
                      
                        y
                        -
                        
                          ρ
                          ⁢
                          
                            
                          
                          ⁢
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                          ⁢
                          
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          θ
                        
                      
                      )
                    
                    2
                  
                  +
                  
                    
                      (
                      
                        
                          z
                          o
                        
                        -
                        
                          ρ
                          ⁢
                          
                            
                          
                          ⁢
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                        
                      
                      )
                    
                    2
                  
                
              
            
            ⁢
            
              

            
            ⁢
            
              O
              z
            
          
          =
          
            
              
                
                  
                    
                      z
                      o
                    
                    -
                    
                      ρ
                      ⁢
                      
                        
                      
                      ⁢
                      cos
                      ⁢
                      
                        
                      
                      ⁢
                      φ
                    
                  
                  ⁢
                  
                    
                  
                
                
                  
                    
                      
                        (
                        
                          x
                          -
                          
                            ρ
                            ⁢
                            
                              
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            φ
                            ⁢
                            
                              
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            θ
                          
                        
                        )
                      
                      2
                    
                    +
                    
                      
                        (
                        
                          y
                          -
                          
                            ρ
                            ⁢
                            
                              
                            
                            ⁢
                            sin
                            ⁢
                            
                              
                            
                            ⁢
                            φ
                            ⁢
                            
                              
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            θ
                          
                        
                        )
                      
                      2
                    
                    +
                    
                      
                        (
                        
                          
                            z
                            o
                          
                          -
                          
                            ρ
                            ⁢
                            
                              
                            
                            ⁢
                            cos
                            ⁢
                            
                              
                            
                            ⁢
                            φ
                          
                        
                        )
                      
                      2
                    
                  
                
              
              ⁢
              
                

              
              ⁢
              I
            
            =
            
              
                (
                
                  
                    I
                    x
                  
                  ,
                  
                    I
                    y
                  
                  ,
                  
                    I
                    z
                  
                
                )
              
              =
              
                (
                
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                  ,
                  
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                  ,
                  
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                  
                
                )
              
            
          
        
      
    
  




The no represents the refractive index of air. The nI represents the refractive index of a freeform surface lens 12. A coordinate of an intersection between the freeform surface 124 and z axis is (0, 0, z0). In one embodiment, z0=10 mm. That is the distance between the intersection and the origin is 10 mm.

The light receiving surface 14 is located on a side of the freeform surface 124, and parallel to a plane defined by x axis and y axis. A coordinate of an intersection between the light receiving surface 14 and z axis is (0, 0, zT). A value of zT relates to a size of the rectangular light spot on the light receiving surface 14. In one embodiment, zT=50 mm.
In the LED freeform surface illumination system 300, the LED point light source 10 and the freeform surface lens 12 are both not limited to be divided into three regions. The light emitting angle φ of the LED point light source 10 can be divided into at least two regions. The freeform surface 124 can be divided into at least two annular regions. Each annular region of the freeform surface 124 is corresponded with one region of the light emitting angle φ of the LED point light source 10. In one embodiment, the light emitting angle φ of the LED point light source 10 is divided into two regions. The freeform surface 124 is divided into two annular regions. Each region of the light emitting angle φ is corresponding with one annular region of the freeform surface 124. The lights of one region of the light emitting angle φ pass the corresponding annular region of the freeform surface 124, to form one rectangular light spot on the light receiving surface 14. The length of the rectangular light spot can be 2Lx, the width of the rectangular light spot can be 2Ly.
Referring to FIG. 8 to FIG. 10, light simulation results of the three annular regions of the freeform surface 124 are obtained by LIGHTTOOLS software. Referring to FIG. 6, when the light emitting angle φ is in a region of 0° to 10°, the lights having the light emitting angle φ of 0° pass the freeform surface 124 to illuminate the center of the rectangular light spot, and the lights having the light emitting angle φ of 10° pass the freeform surface 124 to illuminate the edges of the rectangular light spot. Referring to FIG. 7, to keep the freeform surface 124 continuous, when the light emitting angle φ is in a region of 10° to 20°, the lights having the light emitting angle φ of 10° pass the freeform surface 124 to illuminate the edges of the rectangular spot, and the lights having the light emitting angle φ of 20° pass the freeform surface 124 to illuminate the center of the rectangular light spot. Referring to FIG. 8, to keep the freeform surface 124 continuous, when the light emitting angle φ is in a region of 20° to 30°, the lights having the light emitting angle φ of 20° pass the freeform surface 124 to φ illuminate the center of the rectangular light spot, and the lights having the light emitting angle of 30° pass the freeform surface 124 to illuminate the edges of the rectangular light spot.
Referring to FIG. 11 and FIG. 12, in one embodiment, an energy utilization efficiency of the LED freeform surface illumination system 300 is about 97.2%. An illuminating uniformity of the LED freeform surface illumination system 300 is about 84.3%. The more the annular regions of the freeform surface 124 have, the illuminating uniformity of the freeform surface applied to the LED freeform surface illumination system 300 can be improved.
If the lights emitted from the LED point light source 10 directly reach the light receiving surface 14, a non-uniform light spot would be formed on the light receiving surface 14. In the present method, a freeform surface lens 12 having the freeform surface 124 is arranged between the LED point light source 10 and the light receiving surface 14 to form a uniform rectangular light spot on the light receiving surface 14. The design method of LED freeform surface illumination system and the LED freeform surface illumination system 100 have the following advantages. The freeform surface 124 is divided into at least two annular regions. The light emitting angle of the LED point light source 10 is divided into at least two regions. Each region of the light emitting angle of the LED point light source 10 is corresponding with one annular region of the freeform surface 124. The lights of all the regions pass the corresponding annular regions of the freeform surface 124, to form the same rectangular light spots on a light receiving surface 14. The rectangular light spots formed from the lights of different regions are formed on the same place on the light receiving surface 14. Thus, in the LED freeform surface illumination system, a uniform illumination can be realized on the light receiving surface 14.
Even though numerous characteristics and advantages of certain inventive embodiments have been set out in the foregoing description, together with details of the structures and functions of the embodiments, the disclosure is illustrative only. Changes may be made in detail, especially in matters of arrangement of parts, within the principles of the present disclosure to the full extent indicated by the broad general meaning of the terms in which the appended claims are expressed.
Depending on the embodiment, certain of the steps of methods described may be removed, others may be added, and the sequence of steps may be altered. It is also to be understood that the description and the claims drawn to a method may comprise some indication in reference to certain steps. However, the indication used is only to be viewed for identification purposes and not as a suggestion as to an order for the steps.
The embodiments shown and described above are only examples. Even though numerous characteristics and advantages of the present technology have been set forth in the foregoing description, together with details of the structure and function of the present disclosure, the disclosure is illustrative only, and changes may be made in the detail, especially in matters of shape, size and arrangement of the parts within the principles of the present disclosure up to, and including the full extent established by the broad general meaning of the terms used in the claims. It will therefore be appreciated that the embodiments described above may be modified within the scope of the claims.

## Claims

1. A LED freeform surface illumination system comprising:
a LED point light source located at origin (0, 0, 0) of a three-dimensional Cartesian coordinate system (x, y, z), wherein a light emitting angle φ of the LED point light source is divided into three regions, the three regions are 0≦φ≦φN, φN≦φ≦φS, and φS≦φ≦φT, wherein φT is a largest light emitting angle;
a freeform surface lens having an axis along z axis of the three-dimensional Cartesian coordinate system (x, y, z), the freeform surface lens comprising:
a spherical surface being a light incident surface and having a spherical centre at the origin (0, 0, 0),
a freeform surface opposite to the spherical surface being a light emitting surface, and comprising three annular regions;

a light receiving surface located on a side of the freeform surface and parallel to a plane defined by x axis and y axis;
wherein the freeform surface lens is disposed between the LED point light source and the light receiving surface, each of the three annular regions of the freeform surface corresponds with one of the three regions of the light emitting angle φ, and each of the three annular regions is capable of having lights of corresponded region of the light emitting angle φ passed therethrough to form a rectangular light spot having a length 2Lx and a width 2Ly on the light receiving surface, in φN≦φ≦φS, lights of φ=φN pass the freeform surface to reach edges of the rectangular light spot, lights of φ=φS pass the freeform surface to reach a center of the rectangular light spot.

2. The LED freeform surface illumination system of claim 1, wherein a diameter of the LED point light source 10 is in a range from about 1 mm to about 5 mm.

3. The LED freeform surface illumination system of claim 1, wherein the light emitting angle φ of the LED point light source is divided into three equal regions, three equal proportional regions, or three arbitrary regions.

4. The LED freeform surface illumination system of claim 1, wherein the LED point light source is also located at the origin of a spherical coordinate system (θ, φ, ρ), a system of first order partial differential equations of the freeform surface in the spherical coordinate system (θ, φ, ρ) is expressed as:




  {
  
    
      
        
          
            
              
                
                  ρ
                  φ
                
                =
                
                  
                    ρ
                    ⁡
                    
                      (
                      
                        θ
                        ,
                        φ
                      
                      )
                    
                  
                  ×
                  
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                        ×
                        
                          (
                          
                            
                              
                                n
                                0
                              
                              ×
                              
                                O
                                z
                              
                            
                            -
                            
                              
                                n
                                I
                              
                              ×
                              
                                I
                                z
                              
                            
                          
                          )
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                        ×
                        A
                      
                    
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                        ×
                        
                          (
                          
                            
                              
                                n
                                0
                              
                              ×
                              
                                O
                                z
                              
                            
                            -
                            
                              
                                n
                                I
                              
                              ×
                              
                                I
                                z
                              
                            
                          
                          )
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                        ×
                        A
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                    ρ
                    θ
                  
                  =
                  
                    
                      ρ
                      ⁡
                      
                        (
                        
                          θ
                          ,
                          φ
                        
                        )
                      
                    
                    ×
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                    ×
                    
                      
                        
                          
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                                ×
                                
                                  (
                                  
                                    
                                      
                                        n
                                        0
                                      
                                      ×
                                      
                                        O
                                        x
                                      
                                    
                                    -
                                    
                                      
                                        n
                                        I
                                      
                                      ×
                                      
                                        I
                                        x
                                      
                                    
                                  
                                  )
                                
                              
                              -
                            
                          
                        
                        
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                              ×
                              
                                (
                                
                                  
                                    
                                      n
                                      0
                                    
                                    ×
                                    
                                      O
                                      y
                                    
                                  
                                  -
                                  
                                    
                                      n
                                      I
                                    
                                    ×
                                    
                                      I
                                      y
                                    
                                  
                                
                                )
                              
                            
                          
                        
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                          ×
                          
                            (
                            
                              
                                
                                  n
                                  0
                                
                                ×
                                
                                  O
                                  z
                                
                              
                              -
                              
                                
                                  n
                                  I
                                
                                ×
                                
                                  I
                                  z
                                
                              
                            
                            )
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                          ×
                          A
                        
                      
                    
                  
                
                ;
              
            
          
        
        ⁢
        
          

        
        ⁢
        wherein
        ⁢
        
          

        
        ⁢
        A
      
      =
      
        
          sin
          ⁢
          
            
          
          ⁢
          θ
          ×
          
            (
            
              
                
                  n
                  0
                
                ×
                
                  O
                  y
                
              
              -
              
                
                  n
                  I
                
                ×
                
                  I
                  y
                
              
            
            )
          
        
        +
        
          cos
          ⁢
          
            
          
          ⁢
          θ
          ×
          
            (
            
              
                
                  n
                  0
                
                ×
                
                  O
                  x
                
              
              -
              
                
                  n
                  I
                
                ×
                
                  I
                  x
                
              
            
            )
          
        
      
    
    ;
    
      

    
    ⁢
    
      
        O
        x
      
      =
      
        
          x
          -
          
            ρ
            ⁢
            
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            φ
            ⁢
            
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            θ
          
        
        
          
            
              
                (
                
                  x
                  -
                  
                    ρ
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
                )
              
              2
            
            +
            
              
                (
                
                  y
                  -
                  
                    ρ
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
                )
              
              2
            
            +
            
              
                (
                
                  
                    z
                    o
                  
                  -
                  
                    ρ
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                  
                
                )
              
              2
            
          
        
      
    
    ;
    
      

    
    ⁢
    
      
        O
        y
      
      =
      
        
          y
          -
          
            ρ
            ⁢
            
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            φ
            ⁢
            
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            θ
          
        
        
          
            
              
                (
                
                  x
                  -
                  
                    ρ
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
                )
              
              2
            
            +
            
              
                (
                
                  y
                  -
                  
                    ρ
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
                )
              
              2
            
            +
            
              
                (
                
                  
                    z
                    o
                  
                  -
                  
                    ρ
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                  
                
                )
              
              2
            
          
        
      
    
    ;
    
      

    
    ⁢
    
      
        O
        z
      
      =
      
        
          
            
              z
              o
            
            -
            
              ρ
              ⁢
              
                
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              φ
            
          
          ⁢
          
            
          
        
        
          
            
              
                (
                
                  x
                  -
                  
                    ρ
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
                )
              
              2
            
            +
            
              
                (
                
                  y
                  -
                  
                    ρ
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
                )
              
              2
            
            +
            
              
                (
                
                  
                    z
                    o
                  
                  -
                  
                    ρ
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                  
                
                )
              
              2
            
          
        
      
    
    ;
    
      
        and
        ⁢
        
          

        
        ⁢
        I
      
      =
      
        
          (
          
            
              I
              x
            
            ,
            
              I
              y
            
            ,
            
              I
              z
            
          
          )
        
        =
        
          (
          
            
              sin
              ⁢
              
                
              
              ⁢
              φ
              ⁢
              
                
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              θ
            
            ,
            
              sin
              ⁢
              
                
              
              ⁢
              φ
              ⁢
              
                
              
              ⁢
              sin
              ⁢
              
                
              
              ⁢
              θ
            
            ,
            
              cos
              ⁢
              
                
              
              ⁢
              φ
            
          
          )
        
      
    
    ;
  



no represents a refractive index of air, m represents a refractive index of the freeform surface lens, and a coordinate of an intersection between the freeform surface and z axis is (0, 0, z0).

5. The LED freeform surface illumination system of claim 4, wherein the largest light emitting angle is 30°.

6. The LED freeform surface illumination system of claim 4, wherein z0=10 mm.

7. The LED freeform surface illumination system of claim 4, wherein a coordinate of an intersection between the light receiving surface and z axis is (0, 0, zT), and zT=50 mm.

8. The LED freeform surface illumination system of claim 4, wherein an energy utilization efficiency of the LED freeform surface illumination system is 97.2%.

9. The LED freeform surface illumination system of claim 4, wherein an illuminating uniformity of the LED freeform surface illumination system is 84.3%.

10. The LED freeform surface illumination system of claim 1, wherein φN=10°, φS=20°, and φT=30°.

11. The LED freeform surface illumination system of claim 1, wherein in 0≦φ≦φN lights of φ=0° pass the freeform surface to reach a center of the rectangular light spot, and lights of φ=φN pass the freeform surface to reach edges of the rectangular light spot.

12. The LED freeform surface illumination system of claim 1, wherein in φS≦φ≦φT, lights of φ=φS pass the freeform surface to reach a center of the rectangular light spot, and lights of φ=φT pass the freeform surface to reach edges of the rectangular light spot.

13. A LED freeform surface illumination system comprising:
a LED point light source located at origin (0, 0, 0) of a three-dimensional Cartesian coordinate system (x, y, z), wherein a light emitting angle φ of the LED point light source is divided into at least two regions, wherein φT is a largest light emitting angle of the LED point light source;
a freeform surface lens having an axis along z axis of the three-dimensional Cartesian coordinate system (x, y, z), comprising:
a spherical surface being a light incident surface and having a spherical centre at the origin (0, 0, 0),
a freeform surface opposite to the spherical surface being a light emitting surface, and comprising at least two annular regions;

a light receiving surface located on a side of the freeform surface and parallel to a plane defined by x axis and y axis;
wherein the freeform surface lens is disposed between the LED point light source and the light receiving surface, each of the at least two annular regions of the freeform surface corresponds with one of the at least two regions of the light emitting angle φ, and each of the at least two annular regions is capable of having lights of corresponded region of the light emitting angle φ passed therethrough to form a rectangular light spot having a length 2Lx and a width 2Ly on the light receiving surface, in φN≦φ≦φS, lights of φ=φN pass the freeform surface to reach edges of the rectangular light spot, lights of φ=φS pass the freeform surface to reach a center of the rectangular light spot.

14. A LED freeform surface illumination system comprising:
a LED point light source located at origin (0, 0, 0) of a three-dimensional Cartesian coordinate system (x, y, z), wherein a light emitting angle φ of the LED point light source is divided into three regions, the three regions are 0≦φ≦φN, φN≦φ≦φS, and φS≦φ≦φT, wherein φT is a largest light emitting angle;
a freeform surface lens having an axis along z axis of the three-dimensional Cartesian coordinate system (x, y, z), the freeform surface lens comprising:
a spherical surface being a light incident surface and having a spherical centre at the origin (0, 0, 0),
a freeform surface opposite to the spherical surface being a light emitting surface, and comprising three annular regions;

a light receiving surface located on a side of the freeform surface and parallel to a plane defined by x axis and y axis;

wherein the freeform surface lens is disposed between the LED point light source and the light receiving surface, each of the three annular regions of the freeform surface corresponds with one of the three regions of the light emitting angle φ, and each of the three annular regions is capable of having lights of corresponded region of the light emitting angle φ passed therethrough to form a rectangular light spot having a length 2Lx and a width 2Ly on the light receiving surface, the LED point light source is located at the origin of a spherical coordinate system (θ, φ, ρ), a system of first order partial differential equations of the freeform surface in the spherical coordinate system (θ, φ, ρ) is expressed as:




  {
  
    
      
        
          
            
              
                
                  ρ
                  φ
                
                =
                
                  
                    ρ
                    ⁡
                    
                      (
                      
                        θ
                        ,
                        φ
                      
                      )
                    
                  
                  ×
                  
                    
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                        ×
                        
                          (
                          
                            
                              
                                n
                                0
                              
                              ×
                              
                                O
                                z
                              
                            
                            -
                            
                              
                                n
                                I
                              
                              ×
                              
                                I
                                z
                              
                            
                          
                          )
                        
                      
                      -
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                        ×
                        A
                      
                    
                    
                      
                        cos
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                        ×
                        
                          (
                          
                            
                              
                                n
                                0
                              
                              ×
                              
                                O
                                z
                              
                            
                            -
                            
                              
                                n
                                I
                              
                              ×
                              
                                I
                                z
                              
                            
                          
                          )
                        
                      
                      +
                      
                        sin
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                        ×
                        A
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                    ρ
                    θ
                  
                  =
                  
                    
                      ρ
                      ⁡
                      
                        (
                        
                          θ
                          ,
                          φ
                        
                        )
                      
                    
                    ×
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                    ×
                    
                      
                        
                          
                            
                              
                                sin
                                ⁢
                                
                                  
                                
                                ⁢
                                θ
                                ×
                                
                                  (
                                  
                                    
                                      
                                        n
                                        0
                                      
                                      ×
                                      
                                        O
                                        x
                                      
                                    
                                    -
                                    
                                      
                                        n
                                        I
                                      
                                      ×
                                      
                                        I
                                        x
                                      
                                    
                                  
                                  )
                                
                              
                              -
                            
                          
                        
                        
                          
                            
                              cos
                              ⁢
                              
                                
                              
                              ⁢
                              θ
                              ×
                              
                                (
                                
                                  
                                    
                                      n
                                      0
                                    
                                    ×
                                    
                                      O
                                      y
                                    
                                  
                                  -
                                  
                                    
                                      n
                                      I
                                    
                                    ×
                                    
                                      I
                                      y
                                    
                                  
                                
                                )
                              
                            
                          
                        
                      
                      
                        
                          cos
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                          ×
                          
                            (
                            
                              
                                
                                  n
                                  0
                                
                                ×
                                
                                  O
                                  z
                                
                              
                              -
                              
                                
                                  n
                                  I
                                
                                ×
                                
                                  I
                                  z
                                
                              
                            
                            )
                          
                        
                        +
                        
                          sin
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                          ×
                          A
                        
                      
                    
                  
                
                ;
              
            
          
        
        ⁢
        
          

        
        ⁢
        wherein
        ⁢
        
          

        
        ⁢
        A
      
      =
      
        
          sin
          ⁢
          
            
          
          ⁢
          θ
          ×
          
            (
            
              
                
                  n
                  0
                
                ×
                
                  O
                  y
                
              
              -
              
                
                  n
                  I
                
                ×
                
                  I
                  y
                
              
            
            )
          
        
        +
        
          cos
          ⁢
          
            
          
          ⁢
          θ
          ×
          
            (
            
              
                
                  n
                  0
                
                ×
                
                  O
                  x
                
              
              -
              
                
                  n
                  I
                
                ×
                
                  I
                  x
                
              
            
            )
          
        
      
    
    ;
    
      

    
    ⁢
    
      
        O
        x
      
      =
      
        
          x
          -
          
            ρ
            ⁢
            
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            φ
            ⁢
            
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            θ
          
        
        
          
            
              
                (
                
                  x
                  -
                  
                    ρ
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
                )
              
              2
            
            +
            
              
                (
                
                  y
                  -
                  
                    ρ
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
                )
              
              2
            
            +
            
              
                (
                
                  
                    z
                    o
                  
                  -
                  
                    ρ
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                  
                
                )
              
              2
            
          
        
      
    
    ;
    
      

    
    ⁢
    
      
        O
        y
      
      =
      
        
          y
          -
          
            ρ
            ⁢
            
              
            
            ⁢
            sin
            ⁢
            
              
            
            ⁢
            φ
            ⁢
            
              
            
            ⁢
            cos
            ⁢
            
              
            
            ⁢
            θ
          
        
        
          
            
              
                (
                
                  x
                  -
                  
                    ρ
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
                )
              
              2
            
            +
            
              
                (
                
                  y
                  -
                  
                    ρ
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
                )
              
              2
            
            +
            
              
                (
                
                  
                    z
                    o
                  
                  -
                  
                    ρ
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                  
                
                )
              
              2
            
          
        
      
    
    ;
    
      

    
    ⁢
    
      
        O
        z
      
      =
      
        
          
            
              z
              o
            
            -
            
              ρ
              ⁢
              
                
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              φ
            
          
          ⁢
          
            
          
        
        
          
            
              
                (
                
                  x
                  -
                  
                    ρ
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
                )
              
              2
            
            +
            
              
                (
                
                  y
                  -
                  
                    ρ
                    ⁢
                    
                      
                    
                    ⁢
                    sin
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    θ
                  
                
                )
              
              2
            
            +
            
              
                (
                
                  
                    z
                    o
                  
                  -
                  
                    ρ
                    ⁢
                    
                      
                    
                    ⁢
                    cos
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                  
                
                )
              
              2
            
          
        
      
    
    ;
    
      
        and
        ⁢
        
          

        
        ⁢
        I
      
      =
      
        
          (
          
            
              I
              x
            
            ,
            
              I
              y
            
            ,
            
              I
              z
            
          
          )
        
        =
        
          (
          
            
              sin
              ⁢
              
                
              
              ⁢
              φ
              ⁢
              
                
              
              ⁢
              cos
              ⁢
              
                
              
              ⁢
              θ
            
            ,
            
              sin
              ⁢
              
                
              
              ⁢
              φ
              ⁢
              
                
              
              ⁢
              sin
              ⁢
              
                
              
              ⁢
              θ
            
            ,
            
              cos
              ⁢
              
                
              
              ⁢
              φ
            
          
          )
        
      
    
    ;
  



no represents a refractive index of air, nI represents a refractive index of the freeform surface lens, and a coordinate of an intersection between the freeform surface and z axis is (0, 0, z0).

15. The LED freeform surface illumination system of claim 14, wherein the largest light emitting angle is 30°.

16. The LED freeform surface illumination system of claim 14, wherein φN=10°, φS=20°, and φT=30°.

17. The LED freeform surface illumination system of claim 14, wherein in 0≦φ≦φN, lights of φ=0° pass the freeform surface to reach a center of the rectangular light spot, and lights of φ=φN pass the freeform surface to reach edges of the rectangular light spot.

18. The LED freeform surface illumination system of claim 14, wherein in φN≦φ≦φS, lights of φ=φN pass the freeform surface to reach edges of the rectangular light spot, lights of φ=φS pass the freeform surface to reach a center of the rectangular light spot.

19. The LED freeform surface illumination system of claim 14, wherein in φS≦φ≦φT, lights of φ=φS pass the freeform surface to reach a center of the rectangular light spot, and lights of φ=φT pass the freeform surface to reach edges of the rectangular light spot.