Refraction at a single curvedspherical surface This is the beginning of a sequence of classes whichwill introduce simple and complex lens systems We will start with some terminology which will becomemore important as we get farther along We will also introduce sign conventions which are veryimportant to doing geometrical optics correctly

Refraction at a curved surface Geometrical construction for exact ray Small angle (paraxial) approximation Focal points of single surfacesJanuary 01

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Terminology-object Object is something thatemits light Often symbolized by arrow Each object point emits raysin all directions Usually only base and tip ofarrow are considered Some rays go through opticalsystem, dont

Length of arrow is objectheight

Rays from the object arereferred to as being inobject spaceJanuary 01

raysfrom tipof object

object

Entrance pupilof opticalsystemrays frombase ofobject

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Terminology-image Rays coming from opticalsystem converge to theimage Assuming perfect opticalsystem This is called real image Image can be observed on ascreen

These rays are referred toas being in image space Length of arrow is imageheight Image height / object height= magnificationJanuary 01

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ObjectPoint

Terminology

ab

Optical System

ab

Image Space

cOptical AxisObjectObject SpacespaccyareecaprayImage-s

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 oneray leaving the optical system (a-a, b-b, c-c) For a perfect optical system, all rays leaving an object pointintersect at a single point, the image point

Physically rays in object space are line segments Start at source, end at first surface of optical systemJanuary 01

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

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Terminology-3D considerations

Light leaves the objectgoing in all directions object in vertical plane Rays entering opticalsystem in verticalplane are meridional Rays entering inhorizontal plane aresagittal

Meridional rays

First surface inoptical systemSkew ray

Object

Sagittal rays

For an object on the optical axis all rays are meridional When meridional and sagittal rays form images indifferent positions, the system has astigmatism Any ray which is not meridional (whether sagittal ornot) is called a skew ray skew rays are more difficult to trace or to understandJanuary 01

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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 image1. 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 surface3. Distances between surfaces positive when measured to the right4. Index of refraction positive except negative when light travels to the left5. Angles positive when measured in counterclockwise direction6. Heights positive when above optical axisNote: small font indicates conventions that are only important incatadioptricsystems, i.e. systems which include mirrors LASERS 51January01

Single ray refraction at spherical interfacei

yarlaitiunIt1

n1

V

Refractedrayi

h

C

R

u

t2

n2

All distances, angles and indices are positive except u Snells law relates i and i use geometry to get relation between u, and uReal ray tracing is messy and complicatedWhen angles and beam height, h, are small, theparaxial approximation makes things much easierJanuary 01

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Construction to find refracted rayn1AIndex=n1

parallel

Pn2

BCIndex=n2

Draw incident ray, and a radius to the point,P, where raystrikes surface Mark a point, A, along incident ray a distance n1 from P Mark a point, B, along radius a distance n2 from point A Connect the points A and B The refracted ray is parallel to the line ABJanuary 01

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Rays from one point dont converge toone pointFour rays graphically tracedn1=1n2=1.5

Rays striking higher on surface cross axis closer tosurface undercorrected spherical aberration

Rays not far from optical axis come close to asingle pointJanuary 01

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Paraxial (small angle) rays-Gaussian opticsnu

-nu h/R

h

u1

u2C

s1

n1surface

Vertical scale is greatlyexaggerated

n2

s2index circles

h n1u1 n2u 2From green triangle =Rn2 n1Paraxialn2 n1or n1u1 n2u2 =h refractionR

index circles become lines radius line notequationperpendicular to surfaceSince u1=h/s1, and u2=-h/s2n1 n2 n2 n1 h doesnt+ =appear!!!s1 s2RJanuary 01

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Focal pointsR

Rf2=secondaryfocallength

f1=primaryfocallength

n2>nn1n1Rn2 Rf1 =f2 =n2 n1n2 n1 Rays from an infinitely distant object converge at thesecondary focal point (within the paraxial approximation) set s1= in imaging equation Rays from primary focal point are refracted parallel toaxis, image at infinity set s2= in imaging equation

n1

January 01

n2>n1

1

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Lateral magnificationy1FCs2

s1n1

n2>n1

Objects off the axis also image toa single point in paraxial optics image to a point off the axis inimage space

For extended objects, each pointis imagedJanuary 01

y2

From similar triangles(shaded)y2s2 Rn1s2m==y1s1 + Rn2 s1

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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 becoming from a point on a real object. Optical processing part of brain cant tell differenceJanuary 01

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Real and virtual objects and imagesn2>n1n1Real objectReal image

n1 n2