几何光学101：近轴光线追迹计算

1、Geometrical Optics 101: Paraxial Ray TracingCalculationsRay tracing is the primary method used by optical engineer to determine optical system performance. Ray tracing is the act of manually tracing a ray of light through a system by calculating the angle of refraction/reflection at each surface. Th

2、is met hod is ext remely useful in sys tems with many surfaces, where Gaussian and New to nian imaging equa tions are unsuitable given the degree of complexity.Today, ray t racing soft ware such as ZEMAX or CODE V enable opti cal engineers to quickly simulate the performance of very complicated syst

3、ems. Paraxial ray tracing involves small ray angles and heights. To understand the basic principles of paraxial ray tracing, consider the necessary calculations and ray tracing tables employed in manually tracing rays of light through a system. This will in turn highlight the usefulness of modern co

4、mputing software.PARAXIAL RAY TRACING STEPS: CALCULATING BFL OF A PCXLENSParaxial ray tracing by hand is typically done with the aid of a ray tracing sheet (Figure1). The number of optical lens surfaces is indicated horizontally and the key lens parameters vertically. There are also sections to diff

5、erentiate the marginal and chief ray. Table 1 explains the key optical lens parameters.To illustrate the steps in paraxial ray tracing by hand, consider a plano-convex (PCX) lens. For this example, #49849 25.4mm Diameter x 50.8mm FL lens is used for simplicity. This particular calculation is used to

6、 calculate the back focal length (BFL) of the PCX lens, but it should be noted that ray tracing can be used to calculate a wide variety of system parameters ranging from cardinal points to pupil size and location.SurfaceMARGINAL RAFFigure 1: Sample Ray Tracing SheetTable 1: Optical Lens Parameters f

7、orRay TracingVariableCDescriptionCurvaturetThicknessnIndex of RefractionSurface PoweryuRay HeightRay AngleStep 1: Enter Known ValuesTo begin, enter the known dimensional values of #49-849 into the ray tracing sheet (Figure 2). Surface 0 is the object plane, Surface 1 is the convex surface of the len

8、s, Surface 2 is the plano surface of the lens, and Surface 3 is the image plane (Figure 3).Remember that the curva ture (C) is equivale nt to 1 divided by the radius of curva ture (R). The first thickness value (t) (25mm in this example) is the distance from the object to the first surface of the le

9、ns. This value is arbitrary for incident collimated light (i.e. light parallel to the optical axis of the optical lens). The index of refraction (n) can be approximated as 1 in air and as 1.517 for the NBK7 substrate of the lens.In Figure 2, the red box is the value to be calculated because it is th

10、e distance from the second surface to the point of focus (BFL). The power ()of the individual surfaces is given by the four th line and is calcula ted using Equa tion 1Note： A nega tive sign isadded to this line to make furthercalculationseasier. In this example, Surface 1 is the only surface with p

11、ower as it is the only curved surface in the system.Surface 0Figure 2: Entering Known Lens Parameter Values into Ray Tracing SheetSurface 0Figure 2: Entering Known Lens Parameter Values into Ray Tracing Sheet0 =： (n2 - nJ 匚iSurface01234C0.00t5.00n1.0001.5171.000-4-0.01 0 0t/n253.29 SmOt0Figure 3: Su

12、rfaces of a Plano-Convex (PCX) LensStep 2: Add a Marginal Ray to the SystemThe next step is to add a marginal ray to the system. Since the PCX lens is spherical with a constant radius of curvature and a collimated input beam is used, the ray height (y) is arbitrary. To simplify calculations, use a h

13、eight of 1mm.A collimated beam also means the initial ray angle (u) is 0 degrees. In the ray tracing shee t, nu is simply the angle of the ray multi plied by the refrac tive index of that medium. Both variables are included to make subsequent calculations simpler (Figure 4).25MOLMO1.5171.00025MOLMO1

14、.5171.000Surface01234G.03S1GG.OOG.00Figure 4: Adding a Marginal Ray to the Ray Tracing SheetStep 3: Calculate BFL with Equations and the Ray Tracing SheetRay tracing involves two primary equations in addition to the one for calculating power. Equations 2 - 3 are necessary for any ray tracing calcula

15、tions.where an apostrophe denotes the subsequent surface, angle, thickness, etc. In this example, to find the ray height at Surface 2 (y), take the ray height at Surface 1 (y) and add it to -0.0197 multiplied by 3.296:yr = y + ut.(2.1)=1 + (3-296)(-0.0197) = 0.93505(2.1)Performing this for ray angle

16、 yields the following value. The entire process is repeated until the ray trace is complete (Figure 5).7L II(31)二-0.0197 + (O.9350fi)C-0) = (31)Surface-012341.000001.00400僅那血Surface-012341.000001.00400僅那血ojssoa0.00000-0.01970401970Figure 5: Propagating the Ray through the SystemNow, solve for the BF

17、L by either adjusting the thickness value until the final ray height is 0 (Figure 6) or by backwards calculating the BFL for a ray height of 0. Fo#49-849, the final BFL value is 47.48mm. This is very close to the 47.50mm listed in the lens specifications. The difference is attributed to the rounding

18、 error of using an index of refraction of 1.517 instead of a slightly more accurate value that was used when the lens was initially designed.Surface-QD3B1025LOW-0.0196952380047.4777158t/n1.000401.000000Surface-QD3B1025LOW-0.0196952380047.4777158t/n1.000401.000000.滋 5080.000000.00000皿197031讯Until the

19、 Ray Height althe Image Plane is 0Figure 6: Calculating Back Focal Length of a Plano-Convex (PCX) Lens using a Ray Tracing Shee tDECIPHERING A TWO LENS RAY TRACING SHEETTo completely understand a ray tracing sheet, consider a two lens system consisting of a double-concave (DCV) lens an iris, and ado

20、uble-convex (DCX) lens (Figures 7 - 8). To learn more about DCV and DCX lenses, please rea Understanding Optical Lens GeometriesSurface0123456oiao0.01000.00M010250-0.025055 血25.00J5.00E.OO115.Wn1.K01.517l.flOO1.0001.517LOCO0.00520.00530.0000-0.012f-0.0129t/n5.00003.2?i025.00W25.00005.373i115.48?7MAR

21、GINAL RAr0.00000 OJ17O3 1.193S4 汕测 S.S0M& 9.009b0.00000Q.147410.1 W 0.152260.1 刃册O.Q38I1-0.07801a11.512.510.030.030.0CA/(y t y)2.W32.0002 側Figure 8: Sample Double-Concave (DCV) and Double-Convex (DCX) Ray Tracing SystemThe aperture stop is the limiting aperture and defines how much light is allowed

22、through the system. The aperture stop can be an optical lens surface or an iris, but it is always a physical surface. The ent rance pupil is the image of the aper ture stop when it is imaged through the preceding lens elements into object space. The exit pupil is the image of the aperture stop when

23、it is imaged through the following lens elements into image space.In an optical system, the aperture stop and the pupils are used to define two very important rays. The chief ray is one that begins at the edge of the object and goes through the center of the entrance pupil, exit pupil, and the stop

24、(in other words, it has a height ) of 0 at those locations). The chief ray, therefore, defines the size of the object and image and the locations of the pupils.The marginal ray of an optical system begins on-axis at the object plane. This ray encounters the edge of the pupils and stops and crosses t

25、he axis at the object and image points. The marginal ray, therefore, defines the location of the object and image and the sizes of the pupils.Aperture Stop LocationIf the location of the aperture stop is unknown, a trial ray, known as the pseudo marginal ray, must be propagated through the system. F

26、or an object not at infinity, this ray must begin at the axial position of the object and can have an arbitrary incident angle. For an object at infinity, the ray can begin at an arbitrary height, but must have an incident angle of 0. Once this is accomplished, the aperture stop is simply the surfac

27、e that has the smallest CA/y value, where CA is the surface clear aperture and y is the height of the pseudo marginal ray at that surface.PTtli = rt u +After locating the aperture stop, the pseudo marginal ray can be scaled appropriately to obtain the actual marginal ray (remember the marginal ray s

28、hould touch the edge of the aperture stop). Once the size and location of the aperture stop is known, the marginal ray height is equal to the radius of the stop and the chief ray height is zero at that location. Paraxial ray tracing can then be carried out in both the forward and the reverse directi

29、ons from those points. When doing raytracing in reverse, Equations 4 - 5 are useful. Note the similarities to Equations 2 - Ttli = rt u +(4)Vignetting AnalysisOnce the location and size of the aperture stop is known, use vignetting analysis to see which surfaces will vignette, or cause rays to be bl

30、ocked. Vignetting analysis is accomplished by taking the clear aperture at every surface and dividing it by two. That value is then compared to the heights of the chief and marginal rays at that surface (Equation 6). Equation 6 can be easily reordered to Equation 7. If Equation 7 is true, the surfac

31、e does not vignette.IVI yNotice in the preceding DCV and DCX example how Surface 3 is the aperture stop where the CA/(| + |y|) value is the smalles t among all surfaces. Also, none o ft he surfaces vigne ttebecause all values are greater than or equal to 2.Object/Image Size and LocationObject (Surfa

32、ce 0)Size is 10mm in diameter (twice the chief ray height at Surface 0)Location is 5 mm in front of the first lens (the first thickness value)Image (Surface 6)Size is 18.2554mm in diameter (twice the final chief ray height)Location is 115.4897mm behind the final lens surface (the last thickness valu

33、e)It is important tonote that the Surface 0 chief ray height is positive while the Surface 6 chief ray height is negative. This indicates that the image is inverted.Effective Focal LengthTo solve for the effective focal length (EFL), it is first necessary to trace a pseudo marginal ray through the s

34、ystem for an object at infinity (i.e. the first ray angle will be 0). In Figure 9, an arbitrary initial height of 1 is chosen to simplify calculations. Once this is accomplished, the EFL of the system is given by Equation 8.PSEUDO NURCINAL RAY1ii11.017041 力 7741跚1.W31.4M 戏nu pM00.0070.01M3mo刪o.mFigu

35、re 9: Pseudo Marginal RayField of ViewFOV = 2 (nS )(9)二= 0.364= 20.B56(9)where nu is the first chief ray angle.Lagrange InvariantThe opti cal invaria nt is a useful tool that allows opti cal designers to det ermine various values without having to completely ray trace a system. It is obtained by com

36、paring two rays within a system at any axial point. The optical invariant is constant for any two rays at every point in the system. In other words, if the invariant for a set of two rays is known, ray trace one of the rays and then scale that by the invariant to find the second.The Lagrange Invaria

37、nt is a version of the optical invariant that uses the chief ray and the marginal ray as the two rays of interest. It is solved using Equation 10 and is illustrated in Figure 10.?： -仃0)厂aa=UdesiredUffiisiing%Figure 10: The Lagrange Invariant of Ray TracingREAL-WORLD RAY TRACING AND SOFTWARE ADVANTAG

38、ESWithin paraxial ray tracing, there are several assumptions that introduce erroiinto the calculations. Paraxial ray tracing assumes that the tangent and sine of all angles are equal to the angles themselves (in other words, tan(u) = u and sin(u) = u). This approximation is valid for small angles, b

39、ut can lead to the propagation of error as ray angles increase.Real ray tracing is a method of reducing paraxial error by eliminating the small-angle approximation and by accounting for the sag of each surface to better model the refraction of off-axis rays. As with paraxial ray tracing, real ray tr

40、acing can be done by handith the help of a ray t race shee t. For the sake of brev ity, only the paraxial met hod has been demons trat ed. Ray t racing soft ware such as CODE V and ZEMAX use real ray t racing to model user-inputted optical systems.Ray tracing by hand is a tedious process. Consequent

41、ly, ray tracing software is usually the preferred method of analysis. Figure 11 shows the DCV-DCX system from the section on Deciphering a Two Lens Ray Tracing Shee t. The following ZEMAX screensho t shows a focal length value of 34.699mm - confirming the paraxial calculation previously performed.Surf:TypeComment-RadiusTh1cRuessGlassS