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Lens defects / Aberration a) Chromatic Aberration Since the index varies with wavelength ( dispersion), the focal length of a simple thin lens varies with wavelength. This is a defect. Such a lens will be no good if one wants clear color images in a plane. There are ways to correct this problem (Figure). b) Spherical Aberration All of the off axis rays do not come to a sharp focus. This is corrected by using a nonspherical shape. Optical Instruments Analysis generally involves the laws of reflection and refraction Analysis uses the procedures of geometrical optics To explain certain phenomena, the wave nature of light must be used Optical instruments - Outline Lens Aberration Camera Human Eye Magnifying Glass Optical Microscopes Telescopes Camera The simplest camera is a pinhole camera. There is no lens and an image is formed of an object. There are many examples in nature where a pinhole effect is operating. The spots in the shade under a tree are images of the sun. Picture taken with a pinhole camera. Picture taken with a pinhole camera. Camera The next most complex camera is one with a simple lens – often, like the very cheap one roll of film throw away cameras, the lens is a small piece of plastic. Excellent color corrected and nearly defect free lens cost money and SLRs are examples: Single Lens Reflex Camera. The single lens maybe made up of many separate pieces. Components Light-tight box Converging lens Produces a real image Film behind the lens receives the image Camera Operation Proper focusing leads to sharp images The lens-to-film distance will depend on the object distance and on the focal length of the lens The shutter is a mechanical device that is opened for selected time intervals Most cameras have an aperture of adjustable diameter to further control the intensity of the light reaching the film With a small-diameter aperture, only light from the central portion reaches the film, and spherical aberration is minimized Camera Operation, Intensity Light intensity is a measure of the rate at which energy is received by the film per unit area of the image The intensity of the light reaching the film is proportional to the area of the lens The brightness of the image formed on the film depends on the light intensity Depends on both the focal length and the diameter of the lens Camera, f-numbers The ƒ-number of a camera is the ratio of the focal length of the lens to its diameter ƒ-number = f/D The ƒ-number is often given as a description of the lens “speed” A lens with a low f - number is a “fast” lens Camera, f-numbers, cont. Increasing the setting from one ƒ-number to the next higher value decreases the area of the aperture by a factor of 2 The lowest ƒ-number setting on a camera corresponds to the aperture wide open and the maximum possible lens area in use Simple cameras usually have a fixed focal length and a fixed aperture size, with an ƒ-number of about 11 (large depth of field) Most cameras with variable ƒ-numbers adjust them automatically The Eye The normal eye focuses light and produces a sharp image Essential parts of the eye Cornea – light passes through this transparent structure Aqueous Humor – clear Index of Cornea = 1.38 liquid behind the cornea Index of Aqueous humor = 1.33 Index of Lens = 1.40 Index of Vitreous humor = 1.34 The Eye – Parts, cont. The pupil A variable aperture An opening in the iris The crystalline lens Most of the refraction takes place at the outer surface of the eye Where the cornea is covered with a film of tears The Eyes – Parts, final The iris is the colored portion of the eye It is a muscular diaphragm that controls pupil size The iris regulates the amount of light entering the eye by dilating the pupil in low light conditions and contracting the pupil in high-light conditions The f-number of the eye is from about 2.8 to 16 The Eye – How Does It Work? The cornea-lens system focuses light onto the back surface of the eye This back surface is called the retina The retina contains receptors called rods and cones These structures send impulses via the optic nerve to the brain The brain converts these impulses into our conscious view of the world The Eye – Operation, cont. Rods and Cones Chemically adjust their sensitivity according to the prevailing light conditions The adjustment takes about 15 minutes This phenomena is “getting used to the dark” Accommodation The eye focuses on an object by varying the shape of the crystalline lens through this process An important component is the ciliary muscle, which is situated in a circle around the rim of the lens Thin filaments, called zonules, run from this muscle to the edge of the lens The Eye – Focusing The eye can focus on a distant object The ciliary muscle is relaxed This causes the lens to flatten, increasing its focal length For an object at infinity, the focal length of the eye is equal to the fixed distance between lens and retina This is about 1.7 cm The Eye - Focusing The eye can focus on near objects The ciliary muscles tenses The lens bulges a bit and the focal length decreases The image is focused on the retina The Eye – Near and Far Points The near point is the closest distance for which the lens can accommodate to focus light on the retina Typically at age 10, this is about 18 cm It increases with age The far point of the eye represents the largest distance for which the lens of the relaxed eye can focus light on the retina Normal vision has a far point of infinity Conditions of the Eye Eyes may suffer a mismatch between the focusing power of the lens-cornea system and the length of the eye Eyes may be Farsighted Light rays reach the retina before they converge to form an image Nearsighted Person can focus on nearby objects but not those far away Farsightedness Also called hyperopia The image focuses behind the retina Can usually see far away objects clearly, but not nearby objects Correcting Farsightedness A converging lens placed in front of the eye can correct the condition The lens refracts the incoming rays more toward the principle axis before entering the eye This allows the rays to converge and focus on the retina Nearsightedness Also called myopia In axial myopia the nearsightedness is caused by the lens being too far from the retina In refractive myopia, the lens-cornea system is too powerful for the normal length of the eye Correcting Nearsightedness A diverging lens can be used to correct the condition The lens refracts the rays away from the principle axis before they enter the eye This allows the rays to focus on the retina Presbyopia and Astigmatism Presbyopia (“old-age vision”) is due to a reduction in accommodation ability The cornea and lens do not have sufficient focusing power to bring nearby objects into focus on the retina (=farsightedness) Condition can be corrected with converging lenses In astigmatism, the light from a point source produces a line image on the retina Produced when either the cornea or the lens or both are not perfectly symmetric Diopters (Refractive Power) Optometrists and ophthalmologists usually prescribe lenses measured in diopters The power of a lens in diopters equals the inverse of the focal length in meters P = 1/ƒ E.g., a converging lens of focal length +20 cm has a power of +5.0 diopters Simple Magnifier A simple magnifier consists of a single converging lens This device is used to increase the apparent size of an object The size of an image formed on the retina depends on the angle subtended by the eye The goal is simply to make the image on the retina larger.We want to put the image at the near point of the eye to make it as large on the retina as the eye can accommodate. The Size of a Magnified Image Image at infinity When an object is placed at the near point, the angle subtended is maximum The near point is about 25 cm When the object is placed just inside the focal point of a converging lens, the lens forms a virtual, upright, and enlarged image Image at near point We use converging lenses with f < N Angular Magnification Angular magnification is defined as tanθ ≈ θ Image at infinity Angle with lens h/ f N θ′ M= = = = θ Angle without lens h / N f The angular magnification is a maximum Therefore, . when the image formed by the lens is at the near point of the eye q = - 25 cm Calculated by 1 1 1 = + f d o di 1 1 1 = − do f di N N N = − do f di M = m N Nmax − f di do Image at near point The closer an image is to the eye, the greater the magnification. Since the closest an image can be to the eye and still be in focus is the near point, N N N 25cm = +1 = 1 + M⎛⎜ M== N ⎞⎟ − f −N f ⎝ f do ⎠ Magnification by a Lens With a single lens, it is possible to achieve angular magnification up to about 4 without serious aberrations With one or two additional lenses, which correct the aberrations, a magnification of up to about 20 can be achieved Example: A biologist with a near-point distance of N=26cm examines an insect wing through a magnifying glass whose focal length in f=4.3cm. Find angular magnification when the image produced by the magnifier is a) At infinity b) At the near point THE COMPOUND MICROSCOPE A compound microscope uses two lenses in combination – an objective and an eye piece (ocular) – to produce a magnified image The object to be viewed is placed just outside the focal lenght length of the objective lens We must supply light in order to illuminate the sample. Typically the light source must have heat absorbing filters in place so that the energy of the beam does not boil away the liquid-like biological sample. In advanced scopes, one may also use polarized light in order to see details that ordinary white light will not be able to resolve. Compound Microscope The lenses are separated by a distance L The approach to analyze the image formation is the same as for any two lenses in a row L is much greater than either focal length The image formed by the first lens becomes the object for the second lens The image seen by the eye, I2, is virtual, inverted and very much enlarged Compound Microscope A compound microscope consists of two lenses Gives greater magnification than a single lens The objective lens has a short focal length, ƒo<1 cm The ocular lens (eyepiece) has a focal length, ƒe of a few cm L Magnifications of the Compound Microscope The lateral magnification of the objective is di di m =− ≈− do f objectve The In a typical situation the object is about the focal distance away angular magnification of the eyepiece of the microscope is M eyepiece = N f eyepiece Overall magnification The overall magnification of the microscope is the product of the individual magnifications M total = mobjective M eyepiece = − di N f objectve f eyepiece The magnifications quoted for microscopes assume a near-point distance of 25 cm Other Considerations with a Microscope The ability of an optical microscope to view an object depends on the size of the object relative to the wavelength of the light used to observe it (we will see this in the next chapter interference/diffraction) For example, you could not observe an atom (d ≈ 0.1 nm) with visible light (λ ≈ 500 nm) A compound microscope uses a 75-mm lens as the objective and a 2.0-cm lens as the eyepiece. The specimen will be mounted 122 mm from the objective. Determine (a) the tube length and (b) the total magnification produced by the microscope Telescopes Two fundamental types of telescopes Refracting telescope uses a combination of lenses to form an image (invented Galileo and by Kepler) Reflecting telescope uses a curved mirror and a lens to form an image (invented by Newton) Telescopes can be analyzed by considering them to be two optical elements in a row The image of the first element becomes the object of the second element Refracting Telescope The two lenses are arranged so that the objective forms a real, inverted image of a distance object The image is near the focal point of the eyepiece The two lenses are separated by the distance ƒobj + ƒeye which corresponds to the length of the tube The eyepiece forms an enlarged, inverted image of the first image Note: θ≈hi/fe and θ0 ≈hi/f0 Angular Magnification of a Telescope The angular magnification depends on the focal lengths of the objective and eyepiece Angular Size of image Angular Size of object θ ' hi / f e ƒo m= = = θ hi / f o ƒe Angular magnification is particularly important for observing “nearby” objects (sun, moon,…) Very distance objects still appear as a small point of light Disadvantages of Refracting Telescopes Large diameters are needed to study distant objects Large lenses are difficult and expensive to manufacture The weight of large lenses leads to sagging, which produces aberrations Reflecting Telescope Helps overcome some of the disadvantages of refracting telescopes Replaces the objective lens with a mirror The mirror is often parabolic to overcome spherical aberrations In addition, the light never passes through glass Except the eyepiece Reduced chromatic aberrations Reflecting Telescope, Newtonian Focus The incoming rays are reflected from the mirror and converge toward point A At A, a photographic plate or other detector could be placed A small flat mirror, M, reflects the light toward an opening in the side and passes into an eyepiece A Examples of Telescopes Reflecting Telescopes Largest in the world are 10 m diameter Keck telescopes on Mauna Kea in Hawaii Largest single-mirrored (http://www.astro.caltech.edu/palomar/animations/light path.mov) telescope in US is the 5 m diameter instrument on Mount Palomar in California Refracting Telescopes Largest in the world is Yerkes Observatory in Wisconsin Has a 1 m diameter Hubble Telescope http://hubblesite.org/ A pirate sights a distant ship with a spyglass that gives an angular magnification of 28. If the focal length of the eyepiece is 14 mm, what is the focal length of the objective? The Moon has an angular size of 0.50° when viewed with unaided vision from Earth. Suppose the Moon is viewed through a telescope with an objective whose focal length is 53 cm and an eyepiece whose focal length is 25 mm. What is the angular size of the Moon as seen through this telescope? Resolution The ability of an optical system to distinguish between closely spaced objects is limited due to the wave nature of light Consider two not coherent light sources (like stars) Because of diffraction, the images consist of bright central regions flanked by weaker bright and dark rings (a) Images are resolved and (b) not resolved (sources too close) Rayleigh’s Criterion If the two sources are separated so that their central maxima do not overlap, their images are said to be resolved The limiting condition for resolution is Rayleigh’s Criterion When the central maximum of one image falls on the first minimum of another image, they images are said to be just resolved The images are just resolved when their angular separation satisfies Rayleigh’s criterion Just Resolved If viewed through a slit of width a, and applying Rayleigh’s criterion, the limiting angle of resolution is θ min = λ a For the images to be resolved, the angle subtended by the two sources at the slit must greater than θmin Barely Resolved (Left) and Not Resolved (Right) Resolution with Circular Apertures The diffraction pattern of a circular aperture consists of a central, circular bright region surrounded by progressively fainter rings The limiting angle of resolution depends on the diameter, D, of the aperture θ min = 1.22 λ D QUICK QUIZ 25.1 Suppose you are observing a binary star with a telescope and are having difficulty resolving the two stars. You decide to use a colored filter to help you. Should you choose a blue filter or a red filter? Resolving Power of a Diffraction Grating If λ1 and λ2 are two nearly equal wavelengths between which the grating spectrometer can just barely distinguish, the resolving power, R, of the grating is λ λ R= = λ2 − λ1 Δλ All the wavelengths are nearly the same Resolving Power of a Diffraction Grating, cont A grating with a high resolving power can distinguish small differences in wavelength The resolving power increases with order number R = Nm N is the number of lines illuminated m is the order number All wavelengths are indistinguishable for the zeroth-order maximum m = 0 so R = 0 25.7 Michelson Interferometer The Michelson Interferometer is an optical instrument that has great scientific importance, but is unfamiliar to most people It splits a beam of light into two parts and then recombines them to form an interference pattern It is used to make accurate length measurements Michelson Interferometer, schematic A beam of light provided by a monochromatic source is split into two rays by mirror M One ray is reflected to M1 and the other transmitted to M2 After reflecting, the rays combine to form an interference pattern The glass plate ensures that both rays travel the same distance through glass Measurements with a Michelson Interferometer The interference pattern for the two rays is determined by the difference in their path lengths When M1 is moved a distance of λ/4, successive light and dark fringes are formed This change in a fringe from light to dark is called fringe shift The wavelength can be measured by counting the number of fringe shifts for a measured displacement of M If the wavelength is accurately known, the mirror displacement can be determined to within a fraction of the wavelength