Elementary Optical Systems. Section 13. Magnifiers and Telescopes

Transcription

1 13-1 Elementary Optical Systems Section 13 Magniiers and Telescopes

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

3 Visual Magniication All optical systems that are used with the eye are characterized by a visual magniication or a visual magniying power. While the details o the deinitions o this quantity dier rom instrument to instrument and or dierent applications, the underlying principle remains the same: How much bigger does an object appear to be when viewed through the instrument? The size change is measured as the change in angular subtense o the image produced by the instrument compared to the angular subtense o the object The angular subtense o the object is measured when the object is placed at the optimum viewing condition.

4 Magniiers As an object is brought closer to the eye, the size o the image on the retina increases and the object appears larger. The largest image magniication possible with the unaided eye occurs 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 virtual image o a nearby object or visual observation. The object must be placed inside the ront ocal point o the magniier. h z F h z u M s 13-4 The magniying power MP is deined as (stop at the eye): MP Angular size o the image (with lens) Angular size o the object at the near point um MP dnp 250 mm u U

5 13-5 Magniiers Magniying Power z z z z z With magniier: h z m h z h h z u M h h( z ) z s ( z s) z h h z Unaided eye: u U d h NP u M dnp ( z ) MP d NP u ( z s) U 250 mm Note that as the eye-lens distance decreases, the MP increases. A common assumption is that the lens is located at the eye (s = 0): dnp dnp 250 mm 250 mm MP z z

6 Magniiers Magniying Power Continued dnp dnp 250mm 250mm MP z z The magniication is a unction o both and the image location. The most common deinition o the MP o a magniier assumes that the lens is close to the eye and that the image is presented to a relaxed eye (z' = ). dnp 250mm 10" MP The maximum MP occurs i the image is presented at the near point o the eye: 13-6 dnp 250mm 10" MP1 1 1 z d NP

7 Magniiers Required MP The angular subtense o the image at the eye: u M MPh d MPu NP U MPh 250mm u U h d The human eye has a resolution o about 1 arc minute. A small object must be enlarged to 1 arc minute to be seen. This determines the required 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 up to about 25X are practical; 10X is common.

8 Telescopes Telescopes are aocal or nearly aocal systems used to change the apparent angular size o an object. The image through the telescope subtends an angle ' dierent rom the angle subtended 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 u u Angular Magniication

9 Keplerian Telescope A Keplerian telescope or astronomical telescope consists o two positive lenses separated by the sum o the ocal lengths. The system stop is usually at or near the objective lens. u Objective (Stop) u t = Image at Ininity F F Eye Lens This telescope can be considered to be a combination o an objective plus 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. u hu hu MP u h u u z 13-9 The image presented to the eye is inverted and reverted (rotated by 180 or upside down ). The MP o a Keplerian telescope is negative.

10 Telescopes Magniying Power The telescope is also aocal: 2 m 1 MP m 1 A conundrum to make the scene appear larger, the magnitude o the MP must be greater than one. This implies that the magnitude 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 about the longitudinal magniication. What is important is the angular subtense o the image compared to the object: the angular magniication. m m 2 (in air) u h h mh mh u u 2 L L ml m L m u 1 MP 1 u m h L h u L u z The image gets closer aster than it gets smaller, and the angular subtense increases. The scene appears oreshortened. Telescopes will list only the magnitude o the MP (10X, etc).

11 Angular Magniication Aocal System m 1 MP m m Angular Magniication m m 1/2 1/4 Angular Mag 2

12 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 Gaussian reduction: 1 1 t t t The Gaussian imaging equations 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 unit magniication. Even i m = 1, then all planes are planes o unit magniication. In either case, the principal planes are not deined Similarly, the angular 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 about the ocal points o an aocal system. Since rays parallel to the axis in one optical space never come to ocus in the other optical space, there really are no ocal points.

13 Stops and Pupils In most telescopes, the stop is at the objective lens to minimize the size and cost o this largest element. The objective lens also serves as the entrance pupil. The exit pupil is the image o the stop produced by the eye lens. The distance between the eye lens and the XP is called the eye relie (ER). u u t = F F h u XP z Objective (Stop) Image at Ininity Eye Lens ER

14 Exit Pupil Exit pupil location: z z z ER z z ER z m z 1 1 Exit pupil size: DEP D m D D MP XP EP EP z m z z ER z m 1 A Keplerian telescope produces a real XP to the right o the eye lens. The XP o a visual instrument is also known as the eye circle or the Ramsden circle. Measuring the diameters o the EP and XP allows a simple way o determining the MP o the telescope system: MP D / D EP XP

15 Exit Pupil and The Eye The EP o the eye should be placed at the XP o the telescope to properly couple the two optical systems. I the eye is not at the XP, vignetting can result: XP Ray bundles 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 XP When the eye is in the XP, the entire FOV o the telescope is seen. The eye can rotate to look around within the FOV.

16 Exit Pupil and Eye Relie The XP should be made larger or smaller than the pupil o the eye so that vignetting does not occur 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 pupil translates with eye rotation. A close match o the instrument XP and the eye EP requires precise alignment o the two pupils. Small displacements will change the light level in the image. This is true even i the eye is at the XP. The human eye pupil diameter varies rom 2-8 mm, with a diameter o about 4 mm under ordinary lighting conditions. When the XP o the instrument overills the EP o the eye, the eye becomes the system stop. Larger instruments tend to have large XPs, while compact instruments may have small XP diameters (1-1.5 mm) Suicient eye relie should be provided to allow the eye to access the XP. Hand-held instruments should have mm o eye relie. Microscopes may have as little as 2-3 mm o eye relie. Other systems should have a very long eye relie. For example, a rilescope needs a large ER to avoid problems due to kickback.

17 Telescope and Binocular Speciication Binoculars are a pair o parallel telescopes; one or each eye. The speciication provided on telescopes and binoculars is o the orm AXB (or example 7X35). A MP BObjective Diameter in mm D XP D MP EP

18 Mini Quiz A 5X Keplerian telescope is constructed out 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 pupil or this telescope. [ ] a. ER = 20 mm and DXP = 10 mm [ ] b. ER = 24 mm and DXP = 10 mm [ ] c. ER = 20 mm and DXP = 5 mm [ ] d. ER = 24 mm and DXP = 5 mm 13-18

19 Mini Quiz Solution A 5X Keplerian telescope is constructed out 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 pupil or this telescope. [ ] a. ER = 20 mm and DXP = 10 mm [ ] b. ER = 24 mm and DXP = 10 mm [ ] c. ER = 20 mm and DXP = 5 mm [X] d. ER = 24 mm and DXP = 5 mm t = = 120mm F XP F h z Objective (Stop) D STOP = 25 mm Eye Lens ER

20 13-20 Mini Quiz Solution Page 2 Telescope Design 5X Keplerian: MP 5 5 t 6 120mm 20 mm 100 mm Eye Relie Image Stop Through the Eye Lens: z z z t 120 mm z ER24 mm Exit Pupil Diameter: DEP DSTOP 25 mm m XP z 24 mm z 120 mm 0.2 D m D mm 5 mm XP XP STOP or D EP 25 mm DXP 5 mm MP 5

21 Required Resolution The resolution o the eye is about 1 arc min: 1 arc min 1 arc min MP MP The visual resolution has been miniied into object space. I two objects are separated by, the minimum MP to visually resolve them is: MP MIN 1 arc min For critical work, MPs larger than this value are oten used to minimize visual atigue. There is oten no need to work at the visual resolution limit.

22 Diraction Because light is a wave, it does not ocus to a perect point image. Diraction rom the aperture limits the size o the image spot. For an aberration ree system, an Airy Disk pattern is produced: Pedrotti & Pedrotti The pattern has a bright central core surrounded by rings.

23 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 unction, and /# W is the image space -number Radius r Peak E % Energy in Ring Central Maximum 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 Fourth Zero 4.24 /# 0.0

24 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 useul approximation or determining the best possible resolution or a given -number.

25 Rayleigh Resolution The Rayleigh resolution criterion states that the images o two point objects can be resolved i the peak o one image alls on the irst zero o the other image. The separation equals the radius o the Airy disk: Resolution 1.22 / # The angular resolution is ound by dividing by the ocal length (or image distance): Angular Resolution 1.22 / DEP /# D EP R R 1.22 / # 1.22 D 1.22 / DEP EP

26 Diraction-Based Resolution The angular resolution based on the Rayleigh criterion is Angular Resolution 1.22 / DEP This is the angular separation o two object points or the angular separation o two intermediate image points rom the perspective o the objective lens. Assuming that = 0.55 m, and the D EP is in mm: In the telescope image space or eye space, this angle is magniied by the MP: μm 1mm arc sec 138arc sec D D 1000μm rad 1 D EP EP EP 138 MP MP arc sec D EP When this Rayleigh resolution equals the eye resolution, the maximum useul 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 results as the Airy discs are just being magniied. Magniications several times this limit are used to minimize visual eort. It is easier to view when not at the visual resolution limit. This extra MP is termed Empty Magniication.

27 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 bundle is clipped by the eye lens XP Unvignetted z 50% Vignetted XP z XP z Fully Vignetted (the limit when one ray passes through the XP)

28 Field Stop A ield stop is a physical aperture placed at an intermediate image plane to restrict or limit the system FOV. This aperture serves to limit the ield to a well-corrected or nonvignetted region. This is oten a cosmetic consideration. In a simple Keplerian telescope, the angular ield o view o a telescope is the angular size o the ield stop as seen by the objective lens. The ield stop deines the maximum chie ray angle or the system. A chie ray corresponding to a larger FOV is blocked by the ield stop. XP z Field Stop Here the ield stop limits the FOV to the 50% vignetted FOV.

29 Field Lenses The ield o view o the instrument 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 bundle o rays back towards the axis and into the aperture o the eye lens. Rays rom Objective FIELD - Field Lens Eye Lens Previously Vignetted Rays XP z

30 Field Lens Eye Lens Combination The combination o the ield lens and the eye lens is called an eyepiece. Gaussian reduction easily gives the power o the eyepiece. Assume thin lenses with a separation equal to the ocal length o the eye lens: t t 1/ FIELD FIELD The eyepiece power equals the eye lens power. The MP o the telescope is unchanged. d t The ront principal plane o the eyepiece remains at the eye lens. As a result the magniication o the stop is unchanged. The XP has the same size as without the ield lens d t 2 FIELD FIELD FIELD The ield lens shits the rear principal plane to reduce the original eye relie by d'. Field Lens P d Eye Lens P ER XP z

31 Field Lenses Summary The ield lens does not change the MP o the telescope or the size o the XP. The XP moves closer to the eye lens reducing the eye relie. Maintaining a usable 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 minimize these eects through deocus. - Eye Lens ER XP z FIELD Field Lens P Eye Lens P d ER XP z MP D XP d DEP MP 2 FIELD - ER ER ER d

32 Eyepieces An eyepiece or ocular is the combination o the ield lens and the eye lens. A simple eyepiece does not have a ield lens. A compound eyepiece has both an eye lens and a ield lens. The properties o eyepieces are applicable to other optical instruments such as microscopes. The eyepiece can contain a ield stop at the intermediate image plane to restrict the system FOV. The aperture o a ield lens located at an intermediate image plane serves the unction o a ield stop. Reticles and graticles provide alignment and measurement iducial marks, and they are placed in the intermediate image plane to be superimposed on the image. Since both the reticle and the image are in ocus, reticles must be clean and deect ree The MP o an eyepiece is deined the same as that o a magniier: MP PIECE 250 mm PIECE

33 Compound Eyepieces It is good practice to displace the ield lens rom the intermediate image plane. The two general classiications o compound eyepieces are the Huygens eyepiece and the Ramsden eyepiece. A great number o speciic and historical design variations exist. In both o these conigurations, it is also common to place a ield stop at the intermediate image plane. The intermediate image plane or a Huygens 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 use with reticles as the eyepiece does not change the magniication or size o the intermediate image. This eyepiece has about 50% more eye relie than the Huygens eyepiece. XP z FIELD XP Field Stop z A Kellner eyepiece replaces the singlet eye lens o the Ramsden eyepiece with a doublet or color correction.

34 Galilean Telescope The Galilean telescope uses 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 z 1m 0 The XP is internal or virtual and not accessible to the eye. There is poor coupling between the telescope and the eye, and the FOV o the system is small. There is no intermediate image plane, so it cannot be used with reticles. The Galilean telescope is used or inexpensive systems such as opera glasses. z For a Galilean telescope to be constructed, the negative lens must be stronger than the positive lens.

35 Galilean Telescope Rays rom an o-axis object enter the telescope at and emerge at. Image at Ininity Collimated input light comes out o the telescope at a larger angle. The image appears bigger to the eye. The image in this coniguration is erect (right-side up) MP t = Objective Eye Lens z Eye

36 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 Equivalent Keplerian Telescope: z z t The image in the Keplerian telescope is upside-down and must be corrected with image erecting prisms or a relay lens in order to obtain an erect image.

37 Reversed Galilean Telescope A reversed Galilean telescope provides a miniied erect image (0 < MP < 1). This coniguration is used in door peepholes and many camera viewinders. In these systems, the eye is oten the system stop F F z t Objective Eye Lens m 1 0 MP 1 Erect Image

38 Focusing 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 must still be projected to ininity or viewing with a relaxed eye. The telescope length is adjusted to place the intermediate image at the ront ocal point o the eyepiece (or magniier). z t z 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 ocus without accommodation. The distance rom the intermediate image to the eyepiece can be adjusted to place the image presented to the eye at its ar point Far Point t z z Shown or Myopia O course, both corrections can be combined.

39 Mirror Based Telescopes Newtonian telescope: A positive mirror with a old lat. It is directly analogous 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 analogous relayed Keplerian telescope, an erect image is produced. Ellipse Parabola Parabola Cassegrain telescope: The combination o the positive primary with a negative secondary is the mirror equivalent o a telephoto objective. Hyperbola Parabola The choice o speciic conics used or these telescopes is based on the aberrations and imaging properties o the conic suraces.

40 Telescopes and Imaging Detectors The term telescope has come to mean any system used to view distant objects. In this ormal discussion, a telescope speciically reers to an aocal system used with the eye. Large astronomical telescopes are actually 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 used and the real image produced 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 instrument. Ignoring diraction and aberrations, the resolution is determined by the pixel size or the resolution o the ilm. The pixel size equates to an angular size in object space.

41 Vignetting Example Keplerian Telescope: 5X m = -1/5 = 250 D = 30 a = 15 t = 300 = 50 D = 20 a = 10 u What are the unvignetted, hal-vignetted and ully-vignetted FOVs o this system? Vignetting will occur at the eye lens, since the stop is at the objective. Marginal Ray: 15 u y.06 y y y ut 15 (.06)(300) 3 u u t y my z Chie Ray: u u y 0 y y ut 300u The chie ray height at the eye lens depends on the FOV. The marginal ray is independent o FOV.

42 Vignetting Example Continued At the eye lens Unvignetted: a y y u 3 u U 1.34 Hal Vignetted: a y and a y u u U 1.91 Fully Vignetted: a y y and a y u u U 2.48 a 10 1 U tan u

43 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: y y ut y FIELD FIELD FIELD 15 (.06)(250) 0 y ut 250u 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 It s an image plane! afield u.04 U 2.29 An almost 2X improvement over the base Keplerian telescope results. Note that the power o the ield lens was not required to do this calculation. This power can be used to prevent vignetting at the eye lens and/or to position the exit pupil (eye relie).

44 13-44 Vignetting Example 5X Keplerian Unvignetted: Hal Vignetted: U 1.34 U 1.91

45 13-45 Vignetting Example 5X Keplerian Fully Vignetted: U 2.48 With Field Lens: U 2.29 FIELD d The ER is reduced rom 60 to 35.

46 Vignetting and FOV with Galilean Telescopes In most Galilean telescopes and binoculars, 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 XP). Consider this example o a 3X system: = 150 mm = -50 mm t = 100 mm FOV = ER = 25 mm Pupil Dia = 4 mm FOV = ±1 t ER Eye Pupil D = 30 mm D = 10 mm MP = 3 The hal vignetted object space FOV is about ±1.6 Vignetting occurs at the objective lens.

47 Hal-Vignetted FOV o Galilean Telescopes When looking through a Galilean telescope, it appears that you are looking through a hole well out in ront o the telescope. The hole is the image o the objective lens through the negative eye lens. Vignetting at the objective lens usually limits the FOV o a Galilean telescope. The apparent size 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. u D D z z ER u Eye Pupil - Stop Using the 3X telescope design: = -50 mm z t = = 100 mm D = 30 mm z z t z mm m z / z D 10 mm *Obvious result: the objective is in object space o the telescope and the objective image is in image space. The diameters must be related by the telescope magniication.

48 Hal-Vignetted FOV o Galilean Telescopes The apparent size 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 through the edge o the objective lens. u D D z ER u Eye Pupil - Stop z D mm 10 mm ER 25mm D /2 5 mm utan ER z mm 4.9 MP 1.6 MP This is the hal-vignetted FOV o the telescope. Note that the FOV depends on the value o the Eye Relie or where the telescope is placed relative to the eye.

49 Autocollimator An autocollimator is a widely-used metrology tool or alignment and angle measurement. It is the combination o a collimator and a viewing telescope. The same objective lens is used or both. A point source is placed at the ocal point o the objective, and the resulting collimated beam is relected back into the telescope by the target mirror. The displacement o the returned beam on a measurement reticle measures the mirror tilt. 2 Objective O Point Source Relecting Surace BS Reticle Image displacement in the reticle plane: D 2 O This displacement is independent o the distance to the test surace.

50 Autocollimator Applications The autocollimator is used to measure tilt angles and to align parallel suraces. Other applications include: Surace latness: Roo test (two returned spots): Right angles (with a pentaprism): Resolutions o 0.1 arc sec are quoted or commercial autocollimators. Reerences: Metrology with Autocollimators, Hume Geometrical and Instrumental Optics, Ch 4, D. Goodman (D. Malacara, Ed.)