1.0 MEASUREMENT OF PARAXIAL PROPERTIES OF OPTICAL SYSTEMS

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1 .0 MEASUREMENT OF PARAXIAL PROPERTIES OF OPTICAL SYSTEMS James C. Wyant Optical Sciences Center University of Arizona Tucson, AZ 8572 If we wish to completely characterize the paraxial properties of a lens, it is necessary to measure the exact location of its carinal points, that is, its noal points, focal points, an principal points. For a lens in air the noal points an principal points coincie. For a thin lens, the two principal points coincie at the center of the lens, so the only require measurement is the focal length, while for a thick lens two of the three quantities--focal length, two focal points, or two principal points--must be etermine.. Thin Lenses.. Measurements Base on Image Equation The simplest measurements of the focal length of a thin lens are base on the image equation + p q = f (.) where p is the object istance from the lens (positive if the object is before the lens), q is the image istance from the lens (positive if the image is after the lens), an f is the focal length of the lens. If the lens to be teste has a positive power, a real image can be forme of a pinhole source, an the istances p an q can be measure irectly. When the lens to be teste has a negative power, it shoul be combine with a positive auxiliary lens having sufficient power so that the combination forms a real image. The focal length can then be etermine for the auxiliary lens alone an the combination of lenses. The resultant ata can be use to etermine the power, or focal length, of the negative lens, since (in the thin-lens approximation) the power of the combine lens system is simply the algebraic sum of the powers of the iniviual elements. To obtain a rough measurement of the focal length of a positive lens, an image can be forme of a source locate several focal lengths away from the lens, an the istance between the lens an the resultant image can be taken as the approximate focal length of the lens. The accuracy of this measurement epens, of course, on how far the object is from the lens. Accoring to the Newtonian form of the image equation zz = f 2, (.2) --

2 where z is the istance of the object from the first focal point, an z is the istance to the image from the secon focal point. If the object an image istances are measure in units of the focal length, then they are reciprocals of each other: z =. z (.3) Thus, if the object is 0 "focal lengths" from the first focal point, the image will be locate /0 of a "focal length" from the secon focal point...2 Autocollimation Technique One of the simplest techniques for locating the focal point of a lens is the autocollimation technique illustrate in Fig. -. Light from the source, often a laser, passes through a pinhole an then through the lens whose focal point is to be foun. After passing through the test lens, the beam is reflecte by a plane mirror that is tilte slightly so that the returning beam oes not pass through the pinhole but forms a small spot to one sie of it. The istance between the pinhole an the test lens is then ajuste until the size of this spot is a minimum. The pinhole then lies in the focal plane of the lens. Pinhole Source Test Lens Plane Mirror Fig. -. Autocollimation for locating focal points. The autocollimation techniques can be use to fin the focal length of a negative lens if an auxiliary positive lens is ae as shown in Figure

3 Pinhole Source Auxiliary Lens Plane Mirror Test Lens Focal Length Fig. -2. Use of an auxiliary positive lens to fin the focal length of a negative lens...3 Geneva Gauge A Geneva gauge, illustrate in Fig. -3, can be use to measure the focal length of a thin lens. It consists of three steel prongs, the outer two of which are fixe, an an inner prong that is free to move along its axis, an that is connecte to an inicator gauge through a mechanical linkage. In use, the gauge is presse onto one surface of the lens to be teste, an the surface power is rea irectly from the ial. The proceure is then repeate for the other surface. The net power of the lens, in iopters (reciprocal meters), is the algebraic sum of the two reaings. The focal length, in meters, is the reciprocal of the power. The quantity actually measure by the Geneva gauge is the sag (sagitta) of the surface. The ial of the gauge is calibrate uner the assumption that the refractive inex of the glass is.523. The power of the surface is φ = n = R R (.4) -3-

4 Using this equation, the actual raius of curvature for a surface can be etermine from measurements using a Geneva gauge. A Geneva gauge can be use to etermine the focal length of a lens having a refractive inex other than.523, if the actual inex n lens is known. The true focal length of the lens is foun from the equation f true = f n lens measure. (.5) Generally, measurements mae with a Geneva gauge are accurate to iopter. Before use, a Geneva gauge shoul be presse against a piece of winow glass (or other flat surface) to see if it reas zero power. It is important, when using a Geneva gauge, to make sure that it is perpenicular to the surface uner test...4 Neutralization Test Fig. -3. Geneva Gauge. Another metho for measuring the focal length of a thin lens is the so-calle neutralization test, in which the unknown lens is place in contact with a lens that has a power equal in magnitue, but opposite in sign, to that of the unknown. In this case (in the thin lens approximation), the powers of the two lenses cancel so the total system has zero power. In use, the unknown lens an the known lens are place in contact, an a istant scene is viewe through the resultant combination. The lenses shoul be hel as close to the eye as feasible, to minimize the effects of mismatche power uring the first trials. The total -4-

5 system power is etermine by observing the motion of the scene as the observer moves his hea from sie to sie. If the scene moves in the same irection as his hea motion, the total system power is positive; if the scene moves in the opposite irection to his hea motion, the total system power is negative. The focal length of the unknown lens is equal to the focal length of a known lens of opposite sign, which results in no apparent motion of the scene when the observer moves his hea from sie to sie. During the latter stages of the neutralization test, somewhat increase sensitivity can be obtaine by moving the lenses farther from the eye...5 Focometer A particularly hany instrument for measuring the power of a thin lens is the focometer, sometimes calle the vertex focometer or vertometer. To use a focometer, the unknown lens is place in a holer on the instrument, an a rum on the sie of the instrument is rotate until a target pattern (typically a cross) can be seen in sharp focus through the eyepiece. The power, in iopters, of the unknown lens can then be rea irectly on the rum. The optical system of a focometer is shown in Fig. -4. The instrument consists of a target that can be move back an forth along the optical axis, a collimating lens, a telescope objective an an eyepiece an reticle. A mount is arrange so that the lens to be teste is locate at the secon focal point of the collimating lens. The focometer is ajuste so that the telescope an the eyepiece are focuse on infinity. When no test lens is present, the target will be in focus when it is locate at the first focal point of the collimating lens. When a lens to be teste is inserte in the instrument, it will be necessary to move the target to restore the focus. Since the unknown lens is at the secon focal point of the collimating lens, the Newtonian image equation for this lens becomes z = 2 ( f unknown ) fo (.6) The power φ unknown is linearly proportional to the istance z that the target must be move to restore the focus: φ unknown = 2 z. f o (.7) Because of this linearity, the focometer is a particularly simple an reliable instrument. Typically, the rum is marke in units of 0.25 iopter, an measurements can easily be interpolate to a fraction of this. The lens holer on a focometer is esigne so the back -5-

6 surface of the unknown lens is locate at the secon focal point of the collimating lens. The quantity actually measure by the focometer is the back, or vertex, focal length of the unknown lens; hence the names vertex focometer an vertometer for this instrument. Fig. -4. Optical System of the Focometer..2 Thick Lenses.2. Focal Collimator The focal length an focal points, an hence the principal points, of a thick lens can be measure using an instrument calle a focal collimator. A focal collimator consists of a reticle at the focal point of an achromatic collimating lens, an its use in measuring the focal length of a lens is illustrate in Fig. -5. The focal collimator is illuminate by an extene source, an the lens to be teste is place in the emergent beam. A filar eyepiece inspects the image forme at the focal plane of the test lens. The focal length, f, of the test lens is given by Fo f = A A (-8) where A is the measure size of the image, A is the size of the reticle, an Fo is the focal length of the collimator objective. Note that the focal collimator may be use to measure negative focal lengths as well as positive; one simply uses a microscope objective with a working istance longer than the negative back focus of the lens uner test. In setting up the focal collimator, it is necessary to etermine the collimator constant Fo/A to as high a egree of accuracy as possible. The lens must be turne aroun to etermine the secon focal point position. The principal points are of course locate a focal length istance from the focal points. -6-

7 Reticle Test Lens Image A α α A F o Focal Collimator f Fig. -5. The focal collimator..2.2 Reciprocal Magnification The carinal points of a thick lens can also be measure using the reciprocal magnification metho, which utilizes the fact that for given positions of object an image, there are two possible positions of the lens, as shown in Fig. -6. When the lens is in position, the object of height h forms an image of height h"; when the lens is in position 2, the image height is h. If the istance between lens positions an 2 is, the focal length of the lens is given by f =, (.9) m m where m is the magnitue of the magnification in position. In practice, the magnification is most easily measure by using a trans-illuminate millimeter scale as an object, an a filar eyepiece to inspect the image The reciprocal magnification metho also gives the location of the principal planes since if p is the istance between the object an the first principal plane for position, an q is the corresponing istance between the image an the secon principal plane, p = m an q =. m (-0) The focal points are then a focal istance from the principal points. -7-

8 Reciprocal Magnification Derivation.nb Optics 53 - James C. Wyant Reciprocal Magnification Derivation p q P P 2 h q p h P P 2 = q p f = p + q ; m = q p = magnitue of magnification f = p + pm = p f = m f = i k j + m i k j m + y m { z = m2 m ; p = m m y z; = pm p = p Hm L { m ; q = m

9 Position Position 2 h h h Fig. -6. Reciprocal magnification test. Another way to etermine the principal points from the reciprocal magnification proceure is to combine it with the autocollimation proceure escribe earlier. Let the object be a millimeter scale. As shown in Fig. -7, for fixe positions of the scale an plane mirror, there are three positions of the lens in which the scale will be image back in its own plane. The location of the focal point with respect to the lens is first establishe using the autocollimation proceure. The lens is then move to position (b) in Fig. -7, an the magnification of the scale in the plane of the mirror is measure (it is convenient to cover the mirror with a sheet of graph paper for this measurement). The lens is then move to position (c), an the istance is measure. The focal length of the lens is calculate using Eq. (.9). Since the focal point is separate from the principal point by the focal length, the location of the principal point is then establishe. By reversing the lens, the location of the other focal point an principal point can be foun. -8-

10 Lens being measure Scale (a) (b) (c) Fig. -7. Locating the principal points of a lens by use of reciprocal magnification test an auto-collimation proceure..2.3 Noal-Slie Lens Bench Probably the easiest way to measure the positions of the carinal points of a lens accurately is to use a noal-slie lens bench, which consists of a pivote lens holer equippe with a slie that allows the lens to be shifte axially with respect to the pivotal axis. Thus, by moving the lens forwar or backwar, the lens can be mae to rotate about any esire point. Now note that if the lens is pivote about its secon noal point, as inicate in Fig. -8, the ray emerging from this point (which by efinition emerges from the system parallel to its incoming irection) will coincie with the bench axis (through the noal point). Thus there will be no lateral motion of the image when the lens is rotate about the secon noal point. Once the noal point has been locate in this manner, the lens is then realigne with the collimator axis an the location of the focal point is etermine. Since the noal points an principal points are coincient when a lens is in air, the istance from the noal point to the focal point is the equivalent focal length. -9-

11 N N 2 f θ N N 2 Fig. -8. Rotation about the secon noal point. We see from Fig. -8 that as the lens is rotate about the secon noal point, the istance from the lens to the focus is effectively shortene. If the image of an infinitely istant source is observe, the image moves along a circular arc whose raius of curvature is equal to the focal length of the lens. Most lenses are esigne to form an image on a plane surface, an in orer to inspect the image on such a surface, it is necessary to withraw the observation surface as the lens is rotate by an amount ( sec ) ε = f θ z (.) A lens bench that automatically compensates for this fiel curvature is the Kingslake lens bench, illustrate in Fig. -9. The Kingslake lens bench consists essentially of a noal slie for holing the lens to be teste, a microscope for viewing the image forme by the lens, an a T-bar arrangement that automatically keeps the viewing microscope focuse on a flat fiel. In the T-bar construction of the Kingslake lens bench, the viewing microscope is mounte on a carriage that ries on two longituinal support rails. As the lens is rotate about its noal point, the T-bar swings aroun with it, an the microscope carriage, which is pulle against the T-bar by a tensioning weight, moves back by the proper amount to keep the viewing microscope focuse on a flat fiel. -0-

12 The lens bench is esigne to be use with a pinhole collimator light source. If the bench is to be use for a etaile stuy of the image of the test lens, the quality of the collimator lens shoul be extremely high. In aition, the iameter of the collimator lens must be larger than the iameter of the lens to be teste. This must be especially true if telephoto or retrofocus lenses are to be measure, since with these types of lenses the noal points often lie outsie the lens. The proceure for using the lens bench to fin the location of the carinal points of a lens is straightforwar. Assuming that the bench itself has been previously aligne an calibrate, the lens to be teste shoul be mounte in the holer an the coarse focus ajuste until an image is forme in the plane of the microscope focus. To locate the image, it will generally be necessary to experiment with ifferent centering positions of the lens in its holer. The initial ajustment of the lens in its holer shoul be one with the eyepiece remove from the viewing microscope, an the aerial image of the source through the microscope objective examine irectly, so that the largest possible fiel of view is obtaine. Once the image is locate an focuse in the center of the filar eyepiece fiel, the noal slie shoul be rotate back an forth by a few egrees an the longituinal position of the pivot point ajuste until no motion of the image is observe as the slie is rotate. When this conition is obtaine, the noal point of the lens coincies with the pivot point of the noal slie, an the focal point of the lens is irectly over the axis of the roller on the microscope carriage. Provie the bench is calibrate, the focal length of the lens can then be rea irectly from the scale on the sie of the bench. It is important when measuring the location of the carinal points of a lens not to rotate the noal slie through large angles, as istortion in the lens will then cause a motion of the image even when the lens is pivote about its noal point. In fact, by measuring this image motion, the amount of istortion present in a lens can easily be measure. One of the principal uses of the Kingslake lens bench is the observation an measurement of lens aberrations. In orer for the bench to function properly, a number of ajustments an calibrations must be mae. First, the focus of the microscope must be precisely on the axis of the roller on the microscope carriage. On the Kingslake bench, the microscope is mounte on the carriage so it can be pivote about the axis of the roller. Thus the focus of the microscope can be set on this point by the following proceure: A small object, such as a vertical scratch on a piece of film, is mounte on a stage that permits precision motion both along the axis of the bench, an horizontally perpenicular to this axis (the noal slie itself can be use for such a stage). The microscope is focuse on the scratch. If the scratch is not locate precisely on the axis of the roller, its image will appear to move when the microscope is pivote back an forth about the axis of the roller. If the scratch is isplace along the axis, the image will move from sie to sie, while if the scratch is isplace transversely, the image will appear to move longituinally. Ajustments are mae until the scratch oes not move as the microscope is pivote back an forth; the microscope is given a final focus ajustment an the vernier scales on the transverse an longituinal microscope slies are set to zero. --

13 The optical axis of the microscope must intersect the axis of the pivot on the noal slie. To make this ajustment, the film with the vertical scratch is place on the lens mount, an the coarse focus is ajuste until it is in focus in the microscope (on the Kingslake bench it is necessary to remove the T-bar to make this ajustment). The noal point ajustment an lens holer lateral ajustment are then use to center the scratch by observation of the image motion as the noal slie is rotate back an forth. When the image oes not move, the focus of the microscope lies on the axis of the noal point pivot. The focal length scale shoul then be set to zero. Fig. 9. The Kingslake lens bench (after R. Kingslake, J. Opt. Soc. Am. 22, (932). -2-