Refraction at a single curved spherical surface

Transcription

1 Refraction at a single curved spherical surface This is the beginning of a sequence of classes which will introduce simple and complex lens systems We will start with some terminology which will become more important as we get farther along We will also introduce sign conventions which are very important to doing geometrical optics correctly Refraction at a curved surface Geometrical construction for exact ray Small angle (paraxial) approximation Focal points of single surfaces

2 Terminology-object Object is something that emits light Often symbolized by arrow Each object point emits rays in all directions Usually only base and tip of arrow are considered Some rays go through optical system, don t Length of arrow is object height Rays from the object are referred to as being in object space rays from tip of object object rays from base of object Entrance pupil of optical system

3 Rays coming from optical system converge to the image Assuming perfect optical system This is called real image Image can be observed on a screen These rays are referred to as being in image space Length of arrow is image height Image height / object height = magnification Terminology-image

4 Object Point Optical Axis Object Space a c b Image-space ray Terminology Optical System Image Space Image point Optical system= one or more refracting/reflecting surfaces Often rotationally symmetric, centers of curvature all on one line, the optical axis much more complicated if not symmetric Each ray entering the optical system corresponds to one ray leaving the optical system (a-a, b-b, c-c ) For a perfect optical system, all rays leaving an object point intersect at a single point, the image point Physically rays in object space are line segments Start at source, end at first surface of optical system In optics, we extend the ray in both directions to an infinite line Do the same for image space rays, extend to infinity in both directions b Object-space ray a c

5 Terminology-3D considerations Light leaves the object going in all directions object in vertical plane Rays entering optical system in vertical plane are meridional Rays entering in horizontal plane are sagittal Meridional rays Object Sagittal rays For an object on the optical axis all rays are meridional When meridional and sagittal rays form images in different positions, the system has astigmatism Any ray which is not meridional (whether sagittal or not) is called a skew ray skew rays are more difficult to trace or to understand First surface in optical system Skew ray

6 Geometrical optics sign conventions Very important to be consistent, avoid errors Not universal, many variations Surfaces numbered in the order which light strikes them (not always left to right) Surface zero is object, last surface is image 1. Light travels left to right (for ray-tracing, not in real life!!!!) or exits the system to the right when mirrors are in system 2. Radius positive when the center is to the right of the surface 3. Distances between surfaces positive when measured to the right 4. Index of refraction positive except negative when light travels to the left 5. Angles positive when measured in counterclockwise direction 6. Heights positive when above optical axis Note: small font indicates conventions that are only important in catadioptric systems, i.e. systems which include mirrors

7 Single ray refraction at spherical interface Initial ray t 1 i u V h R Refracted ray i C t 2 u n 1 n 2 All distances, angles and indices are positive except u Snell s law relates i and i use geometry to get relation between u, and u Real ray tracing is messy and complicated When angles and beam height, h, are small, the paraxial approximation makes things much easier

8 Construction to find refracted ray n 1 P parallel A n2 B C Index=n 1 Index=n 2 Draw incident ray, and a radius to the point,p, where ray strikes surface Mark a point, A, along incident ray a distance n 1 from P Mark a point, B, along radius a distance n 2 from point A Connect the points A and B The refracted ray is parallel to the line AB

9 Rays from one point don t converge to one point Four rays graphically traced n 1 =1 n 2 =1.5 Rays striking higher on surface cross axis closer to surface undercorrected spherical aberration Rays not far from optical axis come close to a single point

10 Paraxial (small angle) rays-gaussian optics nu -n u h/r h u 1 u 2 s C 1 s 2 n 1 index circles surface n 2 h n1u From green triangle = R n Vertical scale is greatly exaggerated index circles become lines radius line not perpendicular to surface 1 2 or n2 n1 n1u 1 n2u2 = R Since u 1 =h/s 1, and u 2 =-h/s 2 n1 n2 n2 n1 + = h doesn t s s R appear!!! h 1 2 n2u n 1 2 Paraxial refraction equation

11 Focal points n 2 >n n n 1 2R 1 n n n 2 >n 1 1R 1 f2 = f1 = n2 n n 1 2 n1 Rays from an infinitely distant object converge at the secondary focal point (within the paraxial approximation) set s 1 = in imaging equation Rays from primary focal point are refracted parallel to axis, image at infinity set s 2 = in imaging equation R f 2 =secondary focal length f 1 =primary focal length R

12 Lateral magnification y 1 s 1 s 2 C F y 2 n 1 n 2 >n 1 Objects off the axis also image to a single point in paraxial optics image to a point off the axis in image space For extended objects, each point is imaged m From similar triangles (shaded) y2 s2 R n1s = = y s + R n s

13 How we see For a real object Light emitted from each part is collected by the eye For a real image projected on a screen Screen reflects light from each point, some enters eye For a virtual image Optical system deviates rays so they appear to be coming from a point on a real object. Optical processing part of brain can t tell difference

14 Real and virtual objects and images n 1 n 2 >n 1 n 1 n 2 <n 1 n 1 n 2 <n 1 Real object Real object Virtual object Real image Virtual image Real image Images Real: rays in image space intersect at image Virtual: rays in image space diverge from image point, but don t actually meet at the point Objects Real: rays in object space intersect at object Virtual: rays in object space converge to object point, but don t actually meet at the point Sign convention automatically covers all cases!!

15 Lenses A simple lens consists of two spherical surfaces bounding a homogeneous medium refractive index on both sides of the lens is often but not always the same Ray tracing through a lens (sequential raytracing) trace the ray (either real or paraxial) through the first surface using previous formulae follow it in a straight line till it strikes the second surface trace the ray through the second surface

16 Properties of lenses (paraxial) Second Principal Plane f Secondary focal point Rays from infinity focus at the back focal point The point where the final ray and the incident ray intersect determines the second principal plane Second focal length is the distance from the second principal plane to the second focal point Distance from back surface to focal point called back focal length or BFL

17 Paraxial properties of lenses (cont.) Primary focal point f First Principal Plane Front (or primary) focal point and first principal plane defined similarly distance from primary focal point to first principal plane is the first focal length If the index on both sides of the lens is the same, then first and second focal lengths are the same called effective focal length (EFL)

18 Importance of principal planes Focal points Principal planes Ray from object, parallel to optical axis seems to deflect at 2 nd principal plane; goes through 2 nd focal pt. Ray from object, passing through 1 st focal pt. deflects at 1 st principal plane; emerges parallel to optical axis Image located at intersection of these rays in image space All the refraction appears to take place at the principal planes