Geo OpticsLEOT 330Exam 2Michelle Schroth2. Autocollimation is an optical setup where a collimated beam (of parallel lightrays) leaves an optical system and is reflected back into the same system by aplane mirror. Collimated.It is used for measuring small tilting angles of the mirror, see autocollimator, orfor testing the quality of the optical system or of a part of it. One special applica-tion is to determine the focal length of a diverging lens.3. a. The equivalent power and focal length of the following system is 8 m-1 and12.5 cm.F1sys = f1× f2 ∕f1 + f2 - (d∕n) = 20 cm × 20 cm ∕20 cm + 20 cm - (8 cm∕1) = 12.5 cmF1sys = F2sys = 12.5 cmPequ = P1 + P2 - P1 × P2 × d = 5 m-1 + 5m-1 - (5 m-1) (5 m-1) (0.08 m) = 8 m-13. b. The front vertex power and the front vertex focal length are 13.34 d (m-1)and 7.5 cm. The back vertex power and the back vertex focal length are13.34 d (m-1) and 7.5 cm.
P = 1∕f f1 = 20 cm = .20 mP = 5 m-1 f2 = 20 cm = .20 mP = 5 m-1 d = 8 cm = 0.08 m n=1PFv = P1 + P2∕1 - P2 (d∕n) = 5 m-1 + 5m-1 (0.08 m-1∕1) = 13.34 d (m-1)fFv = 1∕PFv = 1∕13.34 m-1 = 0.075 m = 7.5 cmPFv = PBv = 13.34 d (m-1)fFv = fBv = 1∕13.34 m-1 = 0.075 m = 7.5 cm3. c. The location of the principal planes H1 and H2 are shown below.4. Five common types of optical aberrations are:SphericalWhich is an optical effect observed in an optical device (lens,mirror, etc.) that oc-curs due to the increased refraction of light rays when they strike a lens or a re-flection of light rays when they strike a mirror near its edge, in comparison withthose that strike nearer the center. It signifies a deviation of the device from thenormal operation, i.e., it results in an imperfection of the produced image.
For single lens, spherical aberration can be controlled by bending the lens into itsbest form. Also for multiple lenses, spherical aberrations can be canceled byovercorrecting some elements. The use of symmetric doublets greatly reducespherical aberrations.Chromatic AberrationChromatic aberration or "color fringing" is caused by the camera lens not focus-ing different wavelengths of light onto the exact same focal plane (the focallength for different wavelengths is different) and/or by the lens magnifying differ-ent wavelengths differently. These types of chromatic aberration are referred toas "Longitudinal Chromatic Aberration" and "Lateral Chromatic Aberration" re-spectively and can occur concurrently. The amount of chromatic aberration de-pends on the dispersion of the glass. A lens will not focus different colors in ex-actly the same place because the focal length depends on refraction and the in-dex of refraction for blue light (short wavelengths) is larger than that of red light(long wavelengths). The amount of chromatic aberration depends on the disper-sion of the glass.
One way to minimize this aberration is to use glasses of different dispersion in adoublet or other combination.The use of a strong positive lens made from a low dispersion glass like crownglass coupled with a weaker high dispersion glass like flint glass can correct thechromatic aberration for two colors, e.g., red and blue.Field CurvatureIs where the sharpest focus of the lens is on a curved surface in the image spacerather than a plane. Objects in the center and edges of the field are never in fo-cus simultaneously.
We can correct this aberration by using specially designed objectives. Thesespecially-corrected objectives have been named plan or plano (for flat-field)and are the most common type of objective in use today, providing ocular fieldsranging between 18 and 26 millimeters, which exhibit sharp detail from center toedge.Comatic AberrationsThey are seen mainly with off-axis light fluxes and are most severe when the mi-croscope is out of alignment. With a comatic aberration, the image of a point isfocused at sequentially differing heights producing a series of asymmetrical spotshapes of increasing size that result in a comet-like shape to the Airy pattern.
The distinct shape displayed by images with comatic aberration is a result of re-fraction differences by light rays passing through the various lens zones as theincident angle increases. The severity of comatic aberration is a function of thinlens shape, causing meridional rays passing through the periphery of the lens toarrive at the image plane closer to the axis than do rays passing nearer the axisand closer to the principal ray.When the Off-Axis Distance slider is moved to the far right position, the ray tracediagram shows several skewed light ray paths representing those rays involvedin the aberration. Off-axis light rays often interfere with each other near the focalplane to generate malformed images seen in the microscope. The image pointproduced by a comatic aberration is actually a complicated three-dimensionalasymmetrical diffraction pattern that departs from the classical Airy pattern.What is formed is an elongated structure composed of arcs and ellipsoidal inten-sities that only vaguely resemble the disk-ring arrangement from which the pointspread function evolved.The severity of Coma is heavily dependent upon the shape of the lens. A strong-ly concave positive meniscus lens will demonstrate substantial negative comaticaberration, whereas plano-convex and bi-convex lenses produce comas thatrange from slightly negative to zero. Objects imaged through the convex side ofa plano-convex lens or a convex meniscus lens will have a positive coma.Coma can be corrected by using a combination of lenses that are positionedsymmetrically around a central stop. In order to completely eliminate coma, theAbbe sine condition must be fulfilled: d × n(sinβ) = d × n(sinβ)where d and d are the distances from the optical axis in the image space (primevalues) and object space, n is the refractive index, and β is the viewing angle. A
lens system, such as a microscope condenser or objective, which is free of co-matic aberration is referred to as aplanatic.Astigmatic AberrationAn objective lens for which spherical and coma aberrations have been correctedmay not be able to converge object points off the axis to a point, separating thosepoints into a segment image in a concentric direction and that in a radial direc-tion. This aberration is known as "astigmatic aberration". An objective with anyastigmatic aberration will change the blur orientation of a point image to longitudi-nal or lateral with respect to before or after the focal point.
When an object lies an appreciable distance from the optical axis, the incidentcone of rays will strike the lens asymmetrically, giving rise to the aberrationknown as astigmatism. To describe it, picture the plane which contains both thechief ray, which is the ray which passes through the center of the lens, and theoptical axis. This plane is knows as the meridional, or tangential, plane. Thesagittal plane is defined as the plane containing the chief ray which is also per-pendicular to the tangential plane.When the object is on the optical axis, the cone of rays is symmetrical with re-spect to the spherical surfaces of the lens. In this case the meridional and thesagittal planes are the same, and the ray configurations in all the planes contain-ing the optical axis are identical. In the absence of any spherical aberration, allof the focal lengths are the same and all of the rays arrive at a single focus.When the object is located off axis, the rays come into the lens at an oblique an-gle. Now the configuration of the ray bundle will be different in the meridionaland sagittal planes. Because of this, the focal lengths in these planes will be dif-ferent as well. Basically, the meridional rays are tilted more with respect to thelens than the sagittal rays, and thus have a shorter focal length. Using Fermatsprincipal, we find that the focal length difference depends effectively on the pow-er of the lens and the angle at which the rays are inclined. This is known as theastigmatic difference, and it increases rapidly as the rays become more oblique.
Since there are two distinct focal lengths, the incident conical bundle of rayschanges after being refracted. The cross section of the beam as it leaves thelens is initially circular, but it gradually becomes elliptical with the major axis inthe sagittal plane, until at the tangential focus, FT, the ellipse degenerates into aline (at third order). All the rays from the object traverse this line, which is knownas the primary image. Beyond this point the beams cross section rapidly opensout until it is again circular. At that location the image is a circular blur known asthe circle of least confusion. Moving further from the lens, the beams cross sec-tion again deforms into a line, called the secondary image. This time it is in themeridional plane at the sagittal focus, FS.5. Find the location of the image in the following diagram. If the object is 2 cm inheight, determine the image height. Indicate whether the image is upright of in-verted.Lens 1 do = -20 cm1∕do + 1∕f = 1∕di M = hi∕ho1∕-20 cm + 1∕10 cm = 1∕di = di∕do
20 cm = di = 20 cm∕-20 cm M = -1 (Inverted)hi = (ho) × (m) = (2 cm) × (-1)hi = -2Lens 2 do = 10 cm 1∕do + 1∕f = 1∕di M = hi∕ho1∕10 cm + 1∕-30 cm = = di∕do 15 cm = di = 15 cm∕10 cm M = 1.5 (Upright)Lens 3 is a Mirror. do = 12 cm 1∕do + 1∕f = 1∕di M = hi∕ho1∕12 cm + 1∕10 cm = = di∕do 5.5 cm = di = 5.5 cm ∕12 cm M = 0.458 or .46 (Upright)hi = (ho) × (m) = (-3 cm) ×(0.458)
hi = 1.374 or 1.4 cmThe image is upright and the image height is 1.4 cm.6. See attachment.7. Using the Lens-makers equation, design a double-convex glass lens with a fo-cal length of 60 cm. Assume the refractive index of the glass is 1.50.P = 1∕f = 1∕60 cm = 1∕.6 mP = 1.67 d (m-1)
R2 = -2R1Use the Lens makers formulaP = 1.67 m-1 = (n-1)(1∕R1 - 1∕R2) = (1.50 - 1)(1∕R1 - 1∕-2R1) = (0.5)(1∕R1 - 1∕R1) = (0.5) 1∕R1 [1 + 1∕2] = (0.5) (1∕R1) [1.5] 1.67 m-1 = (.75)∕R1 R1 = .75∕167 m-1 = 0.45 m R1 = 45 cmR2 = -2R1 = -2 × (45 cm)R2 = - 90 cm8. a. Optical dispersion is the phenomenon in which the phase velocity of a wavedepends on its frequency, or alternatively when the group velocity depends onthe frequency. Media having such a property are termed dispersive media. Opti-cal dispersion is sometimes called chromatic dispersion to emphasize its wave-length-dependent nature, or group-velocity dispersion (GVD) to emphasize therole of the group velocity.The most familiar example of chromatic dispersion is a rainbow, in which disper-sion causes the spatial separation of a white light into components of differentwavelengths i.e. different colors.
Chromatic dispersion is especially important to researchers who are designingoptical equipment like cameras, optical microscopes, and telescopes. When alens system is not carefully designed, the system will focus different colors oflight at different spots – and this doesn’t give a very good image! By planning thesystem carefully and using a combination of lenses made out of different materi-als with different indices of refraction, these chromatic aberrations can be greatlyminimized.The effect of chromatic dispersion is also important to people who send shortpulses, which are made up of many different wavelengths, through opticalwaveguides, like optical fiber. Short pulses of EM Rad are used as a way of en-coding data, like voices during a telephone call and the information on this web-site, so that the data can be sent from one place to another. As the pulse travelsin the waveguide, some wavelengths of light travel faster than others. As thepulses travel down the waveguide, they increase in width and overlap with oneanother. If they spread too much, it is difficult to tell where one pulse begins andthe other ends, and this results in information being lost. Researchers who workin the Communications and Fiber Optics fields of optics are developing devicesto combat the effects of dispersion.The dispersion of light by glass prisms is used to construct spectrometers andspectroadiometers. Holographic gratings are also used, as they allow more ac-curate discrimination of wavelengths.
8. b. Principal planes in optical systemsThe two principal planes in a lens system are hypothetical and have the propertythat a ray emerging from the lens appears to have crossed the rear principalplane at the same distance from the axis that that ray appeared to cross the frontprincipal plane, as viewed from the front of the lens. This means that the lens canbe treated as if all of the refraction happened at the principal planes.The principal planes are crucial in defining the optical properties of the system,since it is the distance of the object and image from the front and rear principalplanes that determines the magnification of the system.If the medium surrounding the optical system has a refractive index of 1 (e.g., airor vacuum), then the distance from the principal planes to their corresponding fo-cal points is just the focal length of the system. In the more general case, the dis-tance to the foci is the focal length multiplied by the index of refraction of themedium.For a thin lens in air, the principal planes both lie at the location of the lens. Thepoint where they cross the optical axis is sometimes misleadingly called the opti-cal center of the lens. However, that for a real lens the principal planes do notnecessarily pass through the centre of the lens, and may not lie inside the lens atall.
For a given set of lenses and separations, the principal planes are fixed and donot depend upon the object position. The thin lens equation can be used, but itleaves out the distance between the principal planes. The focal length f is thatgiven by Gullstrand’s equation. The principal planes for a thick lens are illustrat-ed. For practical use, it is often useful to use the front and back vertex powers.