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CALTECH APH 162 - Focusing and Collimating

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447799Email: [email protected] • Web: newport.comOpticsTECHNICAL REFERENCE ANDFUNDAMENTAL APPLICATIONSLENS SELECTION GUIDE SPHERICAL LENSES CYLINDRICAL LENSES KITS OPTICAL SYSTEMS MIRRORSFocusing and CollimatingOptical Ray TracingAn introduction to the use of lenses tosolve optical applications can begin withthe elements of ray tracing. Figure 1demonstrates an elementary ray traceshowing the formation of an image, usingan ideal thin lens. The object height is y1at a distance s1from an ideal thin lens offocal length f. The lens produces animage of height y2at a distance s2on thefar side of the lens.Figure 1By ideal thin lens, we mean a lens whosethickness is sufficiently small that it doesnot contribute to its focal length. In thiscase, the change in the path of a beamgoing through the lens can be consideredto be instantaneous at the center of thelens, as shown in the figure. In theapplications described here, we willassume that we are working with ideallythin lenses. This should be sufficient foran introductory discussion. Considerationof aberrations and thick-lens effects willnot be included here.Three rays are shown in Figure 1. Any twoof these three rays fully determine thesize and position of the image. One rayemanates from the object parallel to theoptical axis of the lens. The lens refractsthis beam through the optical axis at adistance f on the far side of the lens. Asecond ray passes through the opticalaxis at a distance f in front of the lens.This ray is then refracted into a pathparallel to the optical axis on the far sideof the lens. The third ray passes throughthe center of the lens. Since the surfacesof the lens are normal to the optical axisand the lens is very thin, the deflection ofthis ray is negligible as it passes throughthe lens.In addition to the assumption of anideally thin lens, we also work in theparaxial approximation. That is, anglesare small and we can substitute θ inplace of sin θ.MagnificationWe can use basic geometry to look at themagnification of a lens. In Figure 2, wehave the same ray tracing figure withsome particular line segmentshighlighted. The ray through the center ofthe lens and the optical axis intersect atan angle φ. Recall that the oppositeangles of two intersecting lines are equal.Therefore, we have two similar triangles.Taking the ratios of the sides, we have φ= y1/s1= y2/s2This can then be rearranged to givey2/y1= s2/s1= M.The quantity M is the magnification ofthe object by the lens. The magnificationis the ratio of the image size to the objectsize, and it is also the ratio of the imagedistance to the object distance.Figure 2This puts a fundamental limitation on thegeometry of an optics system. If an opticalsystem of a given size is to produce aparticular magnification, then there isonly one lens position that will satisfythat requirement. On the other hand, abig advantage is that one does not needto make a direct measurement of theobject and image sizes to know themagnification; it is determined by thegeometry of the imaging system itself.Gaussian Lens EquationLet’s now go back to our ray tracingdiagram and look at one more set of linesegments. In Figure 3, we look at theoptical axis and the ray through the frontfocus. Again looking at similar trianglessharing a common vertex and, now, angleη, we have y2/f = y1/(s1-f).Rearranging and using our definition ofmagnification, we findy2/y1= s2/s1= f/(s1-f).Rearranging one more time, we finallyarrive at1/f = 1/s1+ 1/s2.This is the Gaussian lens equation. This equation provides the fundamentalrelation between the focal length of thelens and the size of the optical system.A specification of the requiredmagnification and the Gaussian lensequation form a system of two equationswith three unknowns: f, s1, and s2. Theaddition of one final condition will fixthese three variables in an application.This additional condition is often thefocal length of the lens, f, or the size ofthe object to image distance, in whichcase the sum of s1+ s2is given by thesize constraint of the system. In eithercase, all three variables are then fullydetermined.Figure 3Optical InvariantNow we are ready to look at whathappens to an arbitrary ray that passesthrough the optical system. Figure 4shows such a ray. In this figure, we havechosen the maximal ray, that is, the raythat makes the maximal angle with theoptical axis as it leaves the object,passing through the lens at its maximumclear aperture. This choice makes iteasier, of course, to visualize what ishappening in the system, but thismaximal ray is also the one that is ofmost importance in designing anapplication. While the figure is drawn inthis fashion, the choice is completelyarbitrary and the development shownhere is true regardless of which ray isactually chosen.448800Phone: 1-800-222-6440 • Fax: 1-949-253-1680OpticsTECHNICAL REFERENCE ANDFUNDAMENTAL APPLICATIONSLENS SELECTION GUIDESPHERICAL LENSESCYLINDRICAL LENSESKITSOPTICAL SYSTEMSMIRRORSy1fθ1θ2y2y1fθ1θ2y2Figure 4This arbitrary ray goes through the lens ata distance x from the optical axis. If weagain apply some basic geometry, wehave, using our definition of themagnification, θ1= x/s1and θ2= x/s2= (x/s1)(y1/y2). Rearranging, we arrive at y2θ2= y1θ1.This is a fundamental law of optics. Inany optical system comprising onlylenses, the product of the image size andray angle is a constant, or invariant, ofthe system. This is known as the opticalinvariant. The result is valid for anynumber of lenses, as could be verified bytracing the ray through a series of lenses.In some optics textbooks, this is alsocalled the Lagrange Invariant or theSmith-Helmholz Invariant.This is valid in the paraxialapproximation in which we have beenworking. Also, this development assumesperfect, aberration-free lenses. Theaddition of aberrations to ourconsideration would mean thereplacement of the equal sign by agreater-than-or-equal sign in thestatement of the invariant. That is,aberrations could increase the productbut nothing can make it decrease.Application 1: Focusing aCollimated Laser BeamAs a first example, we look at a commonapplication, the focusing of a laser beamto a small spot. The situation is shown inFigure 5. Here we have a laser beam, withradius y1and divergence θ1that isfocused by a lens of focal length f. Fromthe figure, we have θ2= y1/f. The opticalinvariant then tells us that we must havey2= θ1f, because the product of radiusand divergence angle must be constant.Figure 5As a numerical example,


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