Elementary Optical Systems. Section 13. Magnifiers and Telescopes

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1 Elementary Optical Systems Section 13 Magniiers and Telescopes 13-1 Elementary Optical Systems Many optical systems can be nderstood when treated as combinations o thin lenses. Mirror eqivalents exist or many. The goal o this approach is to examine the paraxial properties (image sie and location; entrance and exit ppils; etc.) o a variety o systems. The types o systems examined incldes: Objectives Collimators Magniiers Field lenses Telescopes Eyepieces Microscopes Telecentric systems Relays Illmination systems Scanners 13-2

2 Visal Magniication All optical systems that are sed with the eye are characteried by a visal magniication or a visal magniying power. While the details o the deinitions o this qantity dier rom instrment to instrment and or dierent applications, the nderlying principle remains the same: How mch bigger does an object appear to be when viewed throgh the instrment? The sie change is measred as the change in anglar sbtense o the image prodced by the instrment compared to the anglar sbtense o the object The anglar sbtense o the object is measred when the object is placed at the optimm viewing condition. Magniiers As an object is broght closer to the eye, the sie o the image on the retina increases and the object appears larger. The largest image magniication possible with the naided eye occrs when the object is placed at the near point o the eye, by convention 250 mm or 10 in rom the eye. A magniier is a single lens that provides an enlarged erect virtal image o a nearby object or visal observation. The object mst be placed inside the ront ocal point o the magniier. h F h M s 13-4 The magniying power MP is deined as (stop at the eye): Anglar sie o the image (with lens) MP Anglar sie o the object at the near point M MP d NP 250 mm U

3 Magniiers Magniying Power h m h h h With magniier: Unaided eye: M dnp( ) MP d NP 250 mm ( s) U h h M h h( ) s ( s) h U d NP 13-5 Note that as the eye-lens distance decreases, the MP increases. A common assmption is that the lens is located at the eye (s = 0): dnp dnp 250mm 250mm MP Magniiers Magniying Power Contined dnp dnp 250mm 250mm MP The magniication is a nction o both and the image location. The most common deinition o the MP o a magniier assmes that the lens is close to the eye and that the image is presented to a relaxed eye (' = ). dnp 250mm 10" MP The maximm MP occrs i the image is presented at the near point o the eye: 13-6 dnp 250mm 10" MP1 1 1 d NP

4 Magniiers Reqired MP The anglar sbtense o the image at the eye: M MP MPh MPh d 250mm NP U h U d The hman eye has a resoltion o abot 1 arc minte. A small object mst be enlarged to 1 arc minte to be seen. This determines the reqired MP. MP h d NP dnp NP 250 mm10" 1 arc min.0003 rad 13-7 or MP.075 mm/ h MP.003"/ h Magniiers p to abot 25X are practical; 10X is common. Telescopes Telescopes are aocal or nearly aocal systems sed to change the apparent anglar sie o an object. The image throgh the telescope sbtends an angle ' dierent rom the angle sbtended by the object. The magniying power MP o a telescope is MP MP 1 MP 1 Telescope magniies Telescope miniies 13-8 The angles and ' are oten considered to be paraxial angles. MP Anglar Magniication

5 Keplerian Telescope A Keplerian telescope or astronomical telescope consists o two positive lenses separated by the sm o the ocal lengths. The system stop is sally at or near the objective lens. Objective (Stop) This telescope can be considered to be a combination o an objective pls a magniier. The objective creates an aerial image (a real image in the air) at the common ocal point that is magniied by the eye lens and presented to the relaxed eye at ininity. h h t = Image at Ininity F F Eye Lens MP The image presented to the eye is inverted and reverted (rotated by 180 or pside down ). The MP o a Keplerian telescope is negative. h 13-9 Telescopes Magniying Power The telescope is also aocal: 2 m 1 1 MP m A conndrm to make the scene appear larger, the magnitde o the MP mst be greater than one. This implies that the magnitde o the magniication is less than one. The image is smaller than the object. So how do telescopes work? MP 1 m 1 Don t orget abot the longitdinal magniication. What is important is the anglar sbtense o the image compared to the object: the anglar magniication. m m 2 (in air) h h mh mh 2 L L ml m L m 1 MP 1 m h L h L The image gets closer aster than it gets smaller, and the anglar sbtense increases. The scene appears oreshortened. Telescopes will list only the magnitde o the MP (10X, etc).

6 Anglar Magniication Aocal System m 1 MP Anglar Magniication m m m 1/2 m 1/4 Anglar Mag Aocal Systems Do Not Have Cardinal Points In an aocal system, rays parallel to the optical axis emerge rom the system also parallel to the optical axis so the system has a power = 0 or ininite ocal length. This also ollows rom Gassian redction: 1 1 t t t The Gassian imaging eqations do not apply and the cardinal points are not deined. For an aocal system, the lateral magniication is constant. Unless m = 1, there are no planes o nit magniication. Even i m = 1, then all planes are planes o nit magniication. In either case, the principal planes are not deined Similarly, the anglar magniication o an aocal system is constant. The nodal points cannot be deined. While it is acceptable to state that the ocal points o an aocal system are at ininity, it is better to never talk abot the ocal points o an aocal system. Since rays parallel to the axis in one optical space never come to ocs in the other optical space, there really are no ocal points.

7 Stops and Ppils In most telescopes, the stop is at the objective lens to minimie the sie and cost o this largest element. The objective lens also serves as the entrance ppil. The exit ppil is the image o the stop prodced by the eye lens. The distance between the eye lens and the is called the eye relie (ER). t = F F h Objective (Stop) Image at Ininity ER Eye Lens Exit Ppil Exit ppil location: ER ER m 1 1 Exit ppil sie: DEP D m D D MP EP EP m ER m 1 A Keplerian telescope prodces a real to the right o the eye lens. The o a visal instrment is also known as the eye circle or the Ramsden circle. Measring the diameters o the EP and allows a simple way o determining the MP o the telescope system: MP D / D EP

8 Exit Ppil and The Eye The EP o the eye shold be placed at the o the telescope to properly cople the two optical systems. I the eye is not at the, vignetting can reslt: Ray bndles are shown or dierent FOVs. The eye will see only on-axis (or near-axis) object points. I the eye is displaced laterally, portions o the oaxis ield are seen When the eye is in the, the entire FOV o the telescope is seen. The eye can rotate to look arond within the FOV. Exit Ppil and Eye Relie The shold be made larger or smaller than the ppil o the eye so that vignetting does not occr with head or eye motion. This compensates or eye rotation as the rotation point o the eye is not at the EP o the eye. The ppil translates with eye rotation. A close match o the instrment and the eye EP reqires precise alignment o the two ppils. Small displacements will change the light level in the image. This is tre even i the eye is at the. The hman eye ppil diameter varies rom 2-8 mm, with a diameter o abot 4 mm nder ordinary lighting conditions. When the o the instrment overills the EP o the eye, the eye becomes the system stop. Larger instrments tend to have large s, while compact instrments may have small diameters (1-1.5 mm) Sicient eye relie shold be provided to allow the eye to access the. Hand-held instrments shold have mm o eye relie. Microscopes may have as little as 2-3 mm o eye relie. Other systems shold have a very long eye relie. For example, a rilescope needs a large ER to avoid problems de to kickback.

9 Telescope and Binoclar Speciication Binoclars are a pair o parallel telescopes; one or each eye. The speciication provided on telescopes and binoclars is o the orm AXB (or example 7X35). A MP BObjective Diameter in mm D DEP MP Mini Qi A 5X Keplerian telescope is constrcted ot o two thin lenses. The separation between the two lenses is 120 mm, and the diameter o the objective lens is 25 mm. The system stop is at the objective. Determine the eye relie and the diameter o the exit ppil or this telescope. [ ] a. ER = 20 mm and D = 10 mm [ ] b. ER = 24 mm and D = 10 mm [ ] c. ER = 20 mm and D = 5 mm [ ] d. ER = 24 mm and D = 5 mm 13-18

10 Mini Qi Soltion A 5X Keplerian telescope is constrcted ot o two thin lenses. The separation between the two lenses is 120 mm, and the diameter o the objective lens is 25 mm. The system stop is at the objective. Determine the eye relie and the diameter o the exit ppil or this telescope. [ ] a. ER = 20 mm and D = 10 mm [ ] b. ER = 24 mm and D = 10 mm [ ] c. ER = 20 mm and D = 5 mm [X] d. ER = 24 mm and D = 5 mm t = = 120mm F F h Objective (Stop) D STOP = 25 mm ER Eye Lens Mini Qi Soltion Page 2 Telescope Design 5X Keplerian: MP 5 5 t 6 120mm 20mm 100mm Eye Relie Image Stop Throgh the Eye Lens: t mm ER 24mm Exit Ppil Diameter: D D 25mm EP STOP m 24mm mm D m D mm 5mm STOP or DEP 25mm D 5mm MP 5

11 Reqired Resoltion The resoltion o the eye is abot 1 arc min: 1 arc min 1 arc min MP MP The visal resoltion has been miniied into object space. I two objects are separated by, the minimm MP to visally resolve them is: MP MIN 1 arc min For critical work, MPs larger than this vale are oten sed to minimie visal atige. There is oten no need to work at the visal resoltion limit. Diraction Becase light is a wave, it does not ocs to a perect point image. Diraction rom the apertre limits the sie o the image spot. For an aberration ree system, an Airy Disk pattern is prodced: Pedrotti & Pedrotti The pattern has a bright central core srronded by rings.

12 Airy Disk E E 0 r1 r2 r3 r E r 2 2 J / /# E 1 0 r/ /# where r is the radial coordinate, J 1 is a Bessel nction, and /# W is the image space -nmber Radis r Peak E % Energy in Ring Central Maximm E First Zero 1.22 /# 0.0 First Ring 1.64 /# E Second Zero 2.24 /# 0.0 Second Ring 2.66 /# E Third Zero 3.24 /# 0.0 Third Ring 3.70 /# E Forth Zero 4.24 /# 0.0 Airy Disk Diameter The diameter o the Airy Disk is D = 2.44 /# In visible light, is approximately 0.5 m, and D /# in microns This is a very sel approximation or determining the best possible resoltion or a given -nmber

13 Rayleigh Resoltion The Rayleigh resoltion criterion states that the images o two point objects can be resolved i the peak o one image alls on the irst ero o the other image. The separation eqals the radis o the Airy disk: Resoltion 1.22 / # The anglar resoltion is ond by dividing by the ocal length (or image distance): Anglar Resoltion 1.22 / DEP /# D EP R R 1.22 / # 1.22 D 1.22 / DEP EP Diraction-Based Resoltion The anglar resoltion based on the Rayleigh criterion is Anglar Resoltion 1.22 / DEP This is the anglar separation o two object points or the anglar separation o two intermediate image points rom the perspective o the objective lens. Assming that = 0.55 m, and the D EP is in mm: μm 1mm arc sec 138arc sec D D 1000μm rad 1 D EP EP EP In the telescope image space or eye space, this angle is magniied by the MP: 138 MP MP arc sec DEP When this Rayleigh resoltion eqals the eye resoltion, the maximm sel MP is obtained: 138 MP 1 arc min 60 arc sec arc sec D EP MP 0.43 D D in mm EP EP Beyond this MP, no image improvement reslts as the Airy discs are jst being magniied. Magniications several times this limit are sed to minimie visal eort. It is easier to view when not at the visal resoltion limit. This extra MP is termed Empty Magniication.

14 Telescopes Field o View The FOV o the Keplerian telescope is limited by vignetting at the eye lens. As the FOV or intermediate image height increases, the ray bndle is clipped by the eye lens. 50% Vignetted Unvignetted Flly Vignetted (the limit when one ray passes throgh the ) Field Stop A ield stop is a physical apertre placed at an intermediate image plane to restrict or limit the system FOV. This apertre serves to limit the ield to a well-corrected or nonvignetted region. This is oten a cosmetic consideration. In a simple Keplerian telescope, the anglar ield o view o a telescope is the anglar sie o the ield stop as seen by the objective lens. The ield stop deines the maximm chie ray angle or the system. A chie ray corresponding to a larger FOV is blocked by the ield stop Field Stop Here the ield stop limits the FOV to the 50% vignetted FOV.

15 Field Lenses The ield o view o the instrment can be increased by the addition o a ield lens. This lens is placed at the intermediate image plane, and it bends the chie ray and its bndle o rays back towards the axis and into the apertre o the eye lens. Rays rom Objective FIELD Field Lens - Eye Lens Previosly Vignetted Rays Field Lens Eye Lens Combination The combination o the ield lens and the eye lens is called an eyepiece. Gassian redction easily gives the power o the eyepiece. Assme thin lenses with a separation eqal to the ocal length o the eye lens: t t 1/ FIELD FIELD The eyepiece power eqals the eye lens power. The MP o the telescope is nchanged. d t The ront principal plane o the eyepiece remains at the eye lens. As a reslt the magniication o the stop is nchanged. The has the same sie as withot the ield lens d t 2 FIELD FIELD FIELD The ield lens shits the rear principal plane to redce the original eye relie by d'. Field Lens P d Eye Lens P ER

16 Field Lenses Smmary The ield lens does not change the MP o the telescope or the sie o the. The moves closer to the eye lens redcing the eye relie. Maintaining a sable ER limits the strength o the ield lens and the FOV increase possible or a given eye lens diameter. Since the ield lens is located at an image plane, dirt and imperections on it become part o the image. In practice, the ield lens is oten displaced rom the image plane to minimie these eects throgh deocs. - Eye Lens ER FIELD Field Lens Eye Lens P P d ER - ER MP D d DEP MP 2 FIELD ER ER d Eyepieces An eyepiece or oclar is the combination o the ield lens and the eye lens. A simple eyepiece does not have a ield lens. A compond eyepiece has both an eye lens and a ield lens. The properties o eyepieces are applicable to other optical instrments sch as microscopes. The eyepiece can contain a ield stop at the intermediate image plane to restrict the system FOV. The apertre o a ield lens located at an intermediate image plane serves the nction o a ield stop. Reticles and graticles provide alignment and measrement idcial marks, and they are placed in the intermediate image plane to be sperimposed on the image. Since both the reticle and the image are in ocs, reticles mst be clean and deect ree The MP o an eyepiece is deined the same as that o a magniier: MP PIECE 250 mm PIECE

17 Compond Eyepieces It is good practice to displace the ield lens rom the intermediate image plane. The two general classiications o compond eyepieces are the Hygens eyepiece and the Ramsden eyepiece. A great nmber o speciic and historical design variations exist. In both o these conigrations, it is also common to place a ield stop at the intermediate image plane. The intermediate image plane or a Hygens eyepiece alls between the two elements. FIELD Field Stop The Ramsden eyepiece places the ield lens behind the intermediate image. It is a good choice to se with reticles as the eyepiece does not change the magniication or sie o the intermediate image. This eyepiece has abot 50% more eye relie than the Hygens eyepiece FIELD Field Stop A Kellner eyepiece replaces the singlet eye lens o the Ramsden eyepiece with a doblet or color correction. Galilean Telescope The Galilean telescope ses a positive lens and a negative lens to obtain an erect image and a positive MP (MP > 1). Objective t m 0m1 MP MP 1 Eye Lens F F ER 1m 0 The is internal or virtal and not accessible to the eye. There is poor copling between the telescope and the eye, and the FOV o the system is small. There is no intermediate image plane, so it cannot be sed with reticles. The Galilean telescope is sed or inexpensive systems sch as opera glasses. For a Galilean telescope to be constrcted, the negative lens mst be stronger than the positive lens

18 Galilean Telescope Rays rom an o-axis object enter the telescope at and emerge at. Image at Ininity Objective t = MP Eye Lens Eye Collimated inpt light comes ot o the telescope at a larger angle. The image appears bigger to the eye. The image in this conigration is erect (right-side p). Comparison o Galilean and Keplerian Telescopes For a given MP, the Galilean telescope is shorter than the corresponding Keplerian telescope. Its FOV is also smaller. Galilean Telescope: t Eqivalent Keplerian Telescope: t The image in the Keplerian telescope is pside-down and mst be corrected with image erecting prisms or a relay lens in order to obtain an erect image.

19 Reversed Galilean Telescope A reversed Galilean telescope provides a miniied erect image (0 < MP < 1). This conigration is sed in door peepholes and many camera viewinders. In these systems, the eye is oten the system stop. F F t Objective Eye Lens m 1 0 MP 1 Erect Image Focsing a Telescope While telescopes are deined to be aocal, in practice they oten deviate rom this condition. When the object is not at ininity, the image mst still be projected to ininity or viewing with a relaxed eye. The telescope length is adjsted to place the intermediate image at the ront ocal point o the eyepiece (or magniier). t With reractive error, the ar point o the relaxed eye is no longer at ininity. The ar point is the object distance that is in ocs withot accommodation. The distance rom the intermediate image to the eyepiece can be adjsted to place the image presented to the eye at its ar point Far Point t Shown or Myopia O corse, both corrections can be combined.

20 Mirror Based Telescopes Newtonian telescope: A positive mirror with a old lat. It is directly analogos to a Keplerian telescope. Gregorian telescope: The positive primary mirror is ollowed by a second positive mirror to relay the intermediate image. As with the analogos relayed Keplerian telescope, an erect image is prodced. Ellipse Parabola Parabola Cassegrain telescope: The combination o the positive primary with a negative secondary is the mirror eqivalent o a telephoto objective. Hyperbola Parabola The choice o speciic conics sed or these telescopes is based on the aberrations and imaging properties o the conic sraces. Telescopes and Imaging Detectors The term telescope has come to mean any system sed to view distant objects. In this ormal discssion, a telescope speciically reers to an aocal system sed with the eye. Large astronomical telescopes are actally objectives or cameras where an image array detector is placed at the system ocal point. Two examples o imaging detectors are CDD arrays and photographic ilm. With these detectors, an eye lens is not sed and the real image prodced by the telescope alls on the detector array. The detector is placed at an image plane. Reracting: Relecting: The image can also be relayed to other parts o the instrment. Ignoring diraction and aberrations, the resoltion is determined by the pixel sie or the resoltion o the ilm. The pixel sie eqates to an anglar sie in object space.

21 Vignetting Example Keplerian Telescope: 5X m = -1/5 = 250 D = 30 a = 15 t = 300 = 50 D = 20 a = 10 t What are the nvignetted, hal-vignetted and lly-vignetted FOVs o this system? Vignetting will occr at the eye lens, since the stop is at the objective. 15 Marginal Ray: y.06 y y y t 15 (.06)(300) 3 y my Chie Ray: y 0 y y t 300 The chie ray height at the eye lens depends on the FOV. The marginal ray is independent o FOV. Vignetting Example Contined At the eye lens Unvignetted: Hal Vignetted: a y y U 1.34 a y and a y U a 10 U tan Flly Vignetted: a y y and a y U

22 Vignetting Example Add a Field Lens What happens to the FOV i a ield lens o diameter 20 is added to the 5X Keplerian telescope? tfield 250 At the ield lens: yfield y t FIELD yfield 15 (.06)(250) 0 It s an image plane! y t 250 FIELD FIELD Since y FIELD = 0, all o the vignetting conditions collapse to one. The ield lens is in an image plane, and it serves as a ield stop. Unvignetted Field: a y y FIELD FIELD FIELD U 2.29 afield An almost 2X improvement over the base Keplerian telescope reslts. Note that the power o the ield lens was not reqired to do this calclation. This power can be sed to prevent vignetting at the eye lens and/or to position the exit ppil (eye relie). Vignetting Example 5X Keplerian Unvignetted: U 1.34 Hal Vignetted: U

23 Vignetting Example 5X Keplerian Flly Vignetted: U 2.48 With Field Lens: U FIELD 100 d 25 The ER is redced rom 60 to 35. Vignetting and FOV with Galilean Telescopes In most Galilean telescopes and binoclars, the eye serves as the system stop. The system FOV is limited by vignetting at the objective lens. Note that these systems are not always well deined as there is no prescribed location or the eye relative to the eye lens (no external ). Consider this example o a 3X system: = 150 mm = -50 mm t = 100 mm FOV = ER = 25 mm Ppil Dia = 4 mm FOV = ±1 t ER Eye Ppil D = 30 mm D = 10 mm MP = 3 The hal vignetted object space FOV is abot ±1.6 Vignetting occrs at the objective lens.

24 Hal-Vignetted FOV o Galilean Telescopes When looking throgh a Galilean telescope, it appears that yo are looking throgh a hole well ot in ront o the telescope. The hole is the image o the objective lens throgh the negative eye lens. Vignetting at the objective lens sally limits the FOV o a Galilean telescope. The apparent sie o the image o the objective lens (the hole) as viewed rom the eye determines the hal-vignetted FOV and the chie ray at the eye. D D ER Eye Ppil - Stop Using the 3X telescope design: = -50 mm t = D = 30 mm = 100 mm t mm m / D 10 mm *Obvios reslt: the objective is in object space o the telescope and the objective image is in image space. The diameters mst be related by the telescope magniication. Hal-Vignetted FOV o Galilean Telescopes The apparent sie o the image o the objective lens (the hole) as viewed rom the eye determines the hal-vignetted FOV and the chie ray at the eye. The chie ray in object space also goes throgh the edge o the objective lens. D D ER Eye Ppil - Stop D mm 10 mm ER 25mm D /2 5 mm tan ER mm 4.9 MP 1.6 MP This is the hal-vignetted FOV o the telescope. Note that the FOV depends on the vale o the Eye Relie or where the telescope is placed relative to the eye.

25 Atocollimator An atocollimator is a widely-sed metrology tool or alignment and angle measrement. It is the combination o a collimator and a viewing telescope. The same objective lens is sed or both. A point sorce is placed at the ocal point o the objective, and the reslting collimated beam is relected back into the telescope by the target mirror. The displacement o the retrned beam on a measrement reticle measres the mirror tilt. 2 Objective O Point Sorce Relecting Srace BS Reticle Image displacement in the reticle plane: D 2 O This displacement is independent o the distance to the test srace. Atocollimator Applications The atocollimator is sed to measre tilt angles and to align parallel sraces. Other applications inclde: Srace latness: Roo test (two retrned spots): Right angles (with a pentaprism): Resoltions o 0.1 arc sec are qoted or commercial atocollimators. Reerences: Metrology with Atocollimators, Hme Geometrical and Instrmental Optics, Ch 4, D. Goodman (D. Malacara, Ed.)

Elementary Optical Systems. Section 13. Magnifiers and Telescopes

Elementary Optical Systems. Section 13. Magnifiers and Telescopes 13-1 Elementary Optical Systems Section 13 Magniiers and Telescopes Elementary Optical Systems Many optical systems can be understood when treated as combinations o thin lenses. Mirror equivalents exist