Chapter 1 Geometrical Optics - SPIE

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Chapter 1Geometrical Optics1.1 General CommentsA light wave is an electromagnetic wave, and the wavelength that optics studiesranges from the ultraviolet ( 0.2 mm) to the middle infrared ( 10 mm). Thespatial scales involved in most optical applications are much larger than thelight wavelength. In these cases, a light wave can be approximately described bya bundle of straight optical rays. The science of studying optical rays that travelthrough optical media is called “geometrical optics” and is the most widely usedfield of optics. If the spatial scales involved are not much larger than the lightwavelength, the wave nature of light must be considered; the science of studyingthe wave nature of light is called “wave optics.”The technique to trace an optical ray through various optical media iscalled “sequential raytracing” and is the main way to study geometrical optics.This chapter briefly introduces basic geometrical optics using a sequentialraytracing technique. (Smith1 is a widely cited optical-engineering referencebook, and Hecht2 is a popular optics textbook. Both are recommended here asadditional information sources.)1.2 Snell’s LawWhen an optical ray travels from one optical medium to another that has adifferent refractive index, the ray is split in two by the interface of the twomedia. One ray is reflected back to the first medium. Another ray is refractedand enters the second medium, as shown in Fig. 1.1. The equation describingsuch a refraction is called Snell’s law:3n1 sinðu1 Þ ¼ n2 sinðu2 Þ,(1.1)where u1 is the angle between the incident ray and the normal of the interfaceof the media at the point where the ray hits, u2 is the angle between therefracted ray and the same normal of the interface, and n1 and n2 are therefractive indices of the two media, respectively.1

2Chapter 1Reflected rayn1n21Reflected rayRefracted ray1Refracted ray1Incident rayh22Interface normaland optical axis1Incident rayPlanar interface with n1 n2(a)Symmetric axisof interface andoptical axisn1Local normalof interfaceRn2Spherical interface with n1 n2(b)Figure 1.1 (a) An optical ray is incident on a planar interface of two optical mediawith refractive index n1 and n2, respectively, and n1 n2. (b) An optical ray is incident on aconvex interface of two media with refractive index n1 and n2, respectively, and n1 n2. Thesymmetric axis of the interface is the optical axis. The incident angle varies as the ray heighth varies. For h R, where R is the interface radius of curvature, the interface can beconsidered planar.Figure 1.1(a) illustrates a planar interface. The normal of the interfaceis the normal of the point at which the ray is reflected and refracted, and itis the optical axis. The reflection angle always equals the incident angle, whichis the “reflection law” and will be discussed in Section 1.10.1. For n1 n2,u1 u2, and vice versa, according to Snell’s law.Figure 1.1(b) illustrates a curved interface. The symmetric axis of theinterface is the optical axis. The incident ray height h is the vertical distancebetween the point on the interface where the ray hits and the optical axis. Thefigure shows that the incident angle u1 is a function of h and the interfaceradius of curvature R.Snell’s law is the foundation of geometrical optics, which is why everyoptics book mentions it.1.3 Total Internal ReflectionEquation (1.1) shows that if n1sin(u1)/n2 1, there is no solution for u2because the maximum possible value of sin(u2) is 1. Thus, no ray will berefracted and enter the second medium. Instead, the incident ray will betotally reflected at the interface. This phenomenon is called total internalreflection (TIR).4For example, if medium 1 is N-BK7 glass with an index n1 1.52 andmedium 2 is air with an index n2 1.0, then condition n1sin(u1)/n2 1 leads tou1 41.1 . Any incident ray with incident angle 41.1 will be totally reflected(41.1 is the critical angle in this case). The condition for TIR to happen isn1 n2. A different ratio of n1/n2 would have a different critical angle.

Geometrical Optics3Note that when a ray is incident on an interface of two media with anangle smaller than the TIR angle, the refracted ray will carry most of theenergy of the incident ray, and the reflected ray will carry a relatively muchsmaller portion of the energy. The reflected energy can be calculated usingwave optics theory, as will be discussed in Section 2.3. In many applications,the reflected energy is unwanted and is reduced by applying an antireflection(AR) coating on the interface.1.4 Paraxial ApproximationBefore the invention of computers, tracing a ray using Snell’s law [Eq. (1.1)]through several curved optical interfaces required a lot of calculation anddrawing. Efforts were taken to simplify the situation whenever possible. Whenu1 1, the calculation can be simplified by expanding the sine function inSnell’s law into a series: u2 ¼ sin 1n1sinðu1 Þn2 n11n 33 n1 5sinðu1 Þ þ 1 3 sin ðu1 Þ3 þsin ðu1 Þ5 þ : : :n26 n240 n2 5 3 n11 311 n1 31 3155u ::: þu ::: u u þu u þn2 1 6 1120 16 n2 3 1 6 1120 1 5 3 n1 51 315u ::: þ :::þu u þ40 n2 5 1 6 1120 1 n11 n1 3 n1 31 1 n1 1 n1 3 3 n1 5u 1þu 5 :::: u1 þ þþn26 n2 3 n24 30 n2 3 n2 3 10 n2 5 1 (1.2)The calculation accuracy requirement determines how many terms to keepin Eq. (1.2).“Paraxial approximation”5 means u1 1 so that only the first-order termsof u1, u2 ¼ (n1/n2)u1 must be kept for a fairly accurate calculation. Paraxialapproximation is an ambiguous concept; there is no simple line to determinewhether a u1 value is small enough to qualify for paraxial approximation.Rather, it depends on how accurate the calculation result must be. Figure 1.1(b)shows that h R leads to u1 1. If the terms of the third order or higher of u1are used, the calculation will be quite complex.With the calculation power of modern computers, paraxial approximationis not as important as before. However, paraxial approximation is still usefulfor qualitative and manual analysis, and it is still used in much of theliterature, partially due to tradition.

4Chapter 11.5 Lenses1.5.1 Lens typesLenses are the most commonly used optical component and can be separatedinto two categories: positive and negative.Any lenses with a central thickness larger than their edge thickness arepositive lenses. A positive lens can convergently refract rays that pass throughthe lens, i.e., focus the rays passing through the lens.Any lenses with a central thickness smaller than their edge thickness arenegative lenses. A negative lens can divergently refract the rays passingthrough the lens.Figure 1.2 shows several shapes of positive lenses and negative lenses. Thesymmetric axis of a lens is its optical axis. The functionality of a lens can bedetermined by tracing rays through the lens using Snell’s law.1.5.2 Positive lensesFigure 1.3 shows how raytracing is performed through an equi-convex positivelens to analyze the lens function. The first ray usually traced is parallel tothe optical axis of the lens, traced left to right through the lens, as indicated byFigure 1.2Three shapes of a (a) positive lens and (b) negative lens.RRay 1LRay 2Ray 2FLPLPRRay 1FROptical axisfBLdfLfBRfRPositive lensFigure 1.3 Two rays traced through an equi-convex positive lens. The rays areconvergently refracted. The two cross-points of the rays and the optical axis are the twofocal points of the lens, marked FL and FR, respectively.

Geometrical Optics5Ray 1 in Fig. 1.3. The left surface of the lens convergently refracts the ray, andthe right surface of the lens further convergently refracts the ray. As the raytravels forward, it eventually crosses the optical axis at point FR, which is theright focal point of the lens. The ray is said to be focused.The second ray traced is parallel to the optical axis of the lens and is tracedfrom right to left through the lens, as marked by Ray 2 in Fig. 1.3. The rightsurface of the lens convergently refracts the ray, and the left surface of the lensfurther convergently refracts the ray. The ray eventually crosses the optical axisat point FL, which is the left focal point of the lens. The ray is also focused.For Ray 1, the forward extension of the incident ray and the backwardextension of the exit ray meet at point R. For Ray 2, the forward extension ofthe incident ray and the backward extension of the exit ray meet at point L.Points L and R determine the axial locations of the two principal planes,which are shown by the two vertical dashed lines. The cross-points of the twoprincipal planes and the optical axis are the two principal points, marked PLand PR, respectively, in Fig. 1.3.The axial distance between the left (right) principal plane and the left(right) focal points is the focal length denoted by fL (fR), as marked in Fig. 1.3.fL always equals fR. Lens thickness d is defined as the axial distance betweenthe two vertices of the lens surfaces.The “back focal length” is the axial distance between the left (right) focalpoint and the vertex of the lens left (right) surface, as marked by fBL and fBR,respectively, in Fig. 1.3. fBL equals fBR only when the two surfaces of the lenshave the same shape but opposite orientations. The equi-convex lens shownhere is such a lens.Positive lenses of any shape can focus rays, and their focal lengths are, bydefinition, positive.1.5.3 Negative lensesFigure 1.4 shows how rays are traced through an equal-concave negative lens.Again, the first ray traced is parallel to the optical axis of the lens and is tracedfrom left to right through the lens, as marked by Ray 1 in Fig. 1.4. The leftsurface of the lens divergently refracts the ray, and the right surface of the lensfurther divergently refracts the ray. The ray never crosses the optical axis as ittravels forward. However, a virtual ray that is the leftward extension of theexit Ray 1 can be conceived, as shown by the dashed line in Fig. 1.4. Thisvirtual ray will cross the optical axis at point FL, which is the left focal point ofthe lens. Such a focal point is a virtual focal point; the ray is never actuallyfocused.Similarly, the second ray traced is parallel to the optical axis of the lensand is traced from right to left through the lens, as marked by Ray 2 inFig. 1.4. The right-side surface of the lens divergently refracts the ray, and theleft-side surface of the lens further divergently refracts the ray. The ray

6Chapter 1Ray 1Ray 2RRay 1ConceivedrayLOptical axisPLConceivedrayFLfBLRay 2FRPRdfBRfRfLNegative lensFigure 1.4 Two rays traced through an equal-concave negative lens. The raysare divergently refracted. The two cross-points L and R of the backward extension ofthe exit rays and the optical axis are the two focal points of the lens, marked by FL andFR, respectively. Points L and R determine the axial positions of the two principal pointsPL and PR.never crosses the optical axis. The conceived virtual ray that is extendedrightward from the exit Ray 2 will cross the optical axis at point FR, which isthe right-side focal point of the lens. Such a focal point is also a virtual focalpoint since the ray is never focused.The two principal points and planes of this negative lens can be found byusing the same raytracing technique as described earlier for the equi-convexlens. The focal length and back focal length of a negative lens are measuredthe same way as for the equi-convex positive lens. However, the focal lengthof any negative lens is defined as negative. No negative lens will ever focusrays.1.5.4 Cardinal pointsAny lens has three pairs of cardinal points:6 a pair of principal points, a pairof focal points, and a pair of nodal points. The principal point pair PL and PRand the focal point pair fL and fR were discussed earlier and marked inFigs. 1.3 and 1.4, respectively. This subsection further discusses the opticalmeaning of the principal points and planes.In Figs. 1.3 and 1.4, if the lens is viewed from the right side, the refractedRay 1 appears to be emitted from point R, and PR is the optical position of thelens when viewed from the right side. If the lens is viewed from the left side,the refracted Ray 2 appears to be emitted from point L, and PL is the opticalposition of the lens when viewed from that side.Figure 1.5 shows the positions of the two principal planes for variousshapes of lenses. The principal plane positions of some lenses can be outside

Geometrical OpticsFigure 1.57Positions of the two principal planes for various shapes of lenses.the lenses. If the distance between the two principal points is much smallerthan the focal length, the two principal points can be considered to coincide,and the lens is called a “thin lens.”The raytracings shown in Figs. 1.3 and 1.4 did not consider the effects ofspherical aberrations and are therefore only approximations. The actualpositions of the two principal points and the focal points, and thus the focallength, vary as the incident ray height changes.Figure 1.6 shows accurate raytracing diagrams obtained using the opticaldesign software Zemax. The diagrams illustrate how the position of theprincipal point changes as the incident ray height changes. The positions ofthe principal point in the paraxial approximation (i.e., the incident ray heightapproaches zero), marked by the larger dots in Fig. 1.6, are usually used. Thepositions of the focal points and the value of focal length usually used are alsoparaxial approximation values.1.5.5 Nodal pointsEvery lens has a pair of nodal points. In most cases, including the case shownin Fig. 1.7, the medium at the left and right side of the lens are the same, suchas air. Then the two nodal points coincide with the two principal points.Nodal points have two unique properties:1. A ray aimed at one of the nodal points will be refracted by the lens suchthat it appears to come from the other nodal point and have the sameangle with respect to the optical axis.2. The right (left) focal point position of a lens is not shifted when the lensis rotated about its right (left) nodal point.

8Chapter 1Ray travel directionRay travel directionPLhPROptical axis(b)(a)Ray travel directionRay travel directionhPLPROptical axis(d)(c)Figure 1.6 Zemax-generated accurate raytracing diagram of the principal point positionsas a function of incident ray height h. The principal points are marked by dots. (a) and(b) Rays are traced right and left, respectively, to determine the positions of the two principalpoints for a positive lens with convex-planar surfaces. (c) and (d) Rays are traced right andleft, respectively, to determine the positions of the two principal points for a negative lenswith concave-planar surfaces.Tangential rayMarginal rayObjectChief rayTangential rayPRPLOptical axisFRImagefofiFigure 1.7 An equi-convex positive lens illustrates the nodal points, tangential plane andrays, and the formation of an image. Since the object is placed outside the focal point of thelens, the image formed is a smaller, inverse, and real image. The image is thinner than theobject because the axial magnification (see Section 1.9) is less than 1.

Geometrical Optics9Figure 1.7 illustrates the first property of the nodal points by tracing thechief ray through the lens. (Figures 14.11 and 14.12 in Section 14.6 demonstratethe second property of nodal points with two Zemax-generated, accurateraytracing diagrams.)1.6 Rays and PlanesRaytracing is a geometrical optics technique widely used in the design andanalysis of optical imaging systems. For a given object and lens, the approximate location, size, and orientation of the image formed by the lens can befound by tracing a few rays from the object through the lens. Although raytracing is now mainly performed by computers and optical software, familiaritywith raytracing is still helpful.To effectively trace rays, some special rays and planes must first bedefined. Each of them has a special name and meaning. Figures 1.7 and 1.8use the simplest one-lens example to illustrate these rays and planes.1.6.1 Tangential planes and tangential raysFor a given lens and object, two orthogonal planes and several types of raysare defined. The tangential plane is defined by the optical axis and the objectpoint from which the ray originated. In Fig. 1.7, the tangential plane is theplane of the page. A tangential plane is also called a meridional plane.A ray that travels in the tangential plane is a tangential ray or meridionalray. All four rays shown in Fig. 1.7 are tangential rays or meridional rays.There are different types of tangential rays, such as chief and marginal rays.The chief ray travels from the top of the object through the center of theaperture stop (see Section 1.7). In Fig. 1.7, the lens aperture is the aperture stop.A chief ray is also called a principal ray and is frequently used in raytracing.ImageOptical axisOptical axisObjectLensFigure 1.8 Sagittal plane and sagittal rays. The plane marked by the thin vertical lines isthe tangential plane. Dashed lines indicate tangential rays. The shaded plane is the sagittalplane. All of the rays in the sagittal plane are sagittal rays, drawn with solid lines. Theintersection line of the two planes is the chief ray, denoted by the thick solid line.

10Chapter 1A marginal ray travels from the point where the object and the optical axiscross the edge of the aperture stop. In Fig. 1.7, the lens edge is the aperturestop edge. Tracing at least two tangential rays from an object point through alens can determine the location of the image point, as shown in the figure.1.6.2 Sagittal planes, sagittal rays, and skew raysThere are an infinite number of planes that are perpendicular to the tangentialplane. Among these planes, only one plane contains the chief ray; such a planeis called the sagittal plane, as shown in Fig. 1.8.Rays traveling in the sagittal plane are sagittal rays. The chief ray is theintersecting line of the tangential and sagittal planes. Technically, it is also asagittal ray, but it is usually considered as a tangential ray. Sagittal rays arealso called transverse rays. Sagittal rays are rarely used in manual raytracingbecause it is difficult to conceive and draw rays perpendicular to the plane ofthe page. However, modern computers and optical software can easily tracesagittal rays.Skew rays are those rays that neither travel in the tangential plane norcross the optical axis anywhere, and they are not parallel to the optical axis.Sagittal rays, except the chief ray, are special skew rays.1.7 Stops and Pupils1.7.1 DefinitionsStops are important optical components in an optical system. Pupils are theimages of stops. Most image lenses have an aperture stop and a field stop. Theaperture stop sets the largest cone angle the lens imposes on the object. The fieldstop sets the largest field angle the lens can see. These stops can significantlyaffect the lens characteristics and should be understood. The stops can be eitherlens aperture edges or some other structures.Most image lenses also have an entrance pupil and an exit pupil. These twopupils are the images of the aperture stop formed by the lens at the object side andimage side, respectively. Manually locating the stops and pupils and determiningtheir sizes can be complex, whereas computer and optical design software can easilyperform these tasks. Three examples are included here to explain the process.1.7.2 Example 1: a schematic microscopeFigure 1.9 includes a microscope that consists of an objective and an eyepiece toillustrate the concepts of an aperture stop, field stop, entrance pupil, and exitpupil. In Fig. 1.9(a), the aperture of the objective restrains the cone angle of themicroscope imposing on the object and is the aperture stop. The aperture of theeyepiece restrains the maximum allowed field angle. For example, the rays fromthe solid square on the object plane are completely blocked by the edge of the

Geometrical Optics11Rays areblockedAperture stop andentrance pupilField st

called “sequential raytracing” and is the main way to study geometrical optics. This chapter briefly introduces basic geometrical optics using a sequential raytracing technique. (Smith1 is a widely cited optical-engineering r

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