Chapter 25 Optical Instruments

Analysis generally involves the laws of reflection and refraction Analysis uses the procedures of geometric optics To explain certain phenomena, the wave nature of light must be used

The Camera The single-lens photographic camera is an optical instrument 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 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 liquid behind the cornea

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 – Operation 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 The zonules tighten 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, cont The eye can focus on near objects The ciliary muscles tenses This relaxes the zonules 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 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 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 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

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 Size of a Magnified Image When an object is placed at the near point, the angle subtended is a maximum The near point is about 25 cm When the object is placed near the focal point of a converging lens, the lens forms a virtual, upright, and enlarged image

Angular Magnification Angular magnification is defined as The angular magnification is at a maximum when the image formed by the lens is at the near point of the eye q = - 25 cm Calculated by

Magnification by a Lens With a single lens, it is possible to achieve angular magnification up to about 4 without serious aberrations With multiple lenses, magnifications of up to about 20 can be achieved The multiple lenses can correct for aberrations

Compound Microscope A compound microscope consists of two lenses Gives greater magnification than a single lens The objective lens has a short focal length, ƒ o <1 cm The ocular lens (eyepiece) has a focal length, ƒ e, of a few cm

Compound Microscope, cont The lenses are separated by a distance L L is much greater than either focal length The approach to analysis is the same as for any two lenses in a row The image formed by the first lens becomes the object for the second lens The image seen by the eye, I 2, is virtual, inverted and very much enlarged

Magnifications of the Compound Microscope The lateral magnification of the microscope is The angular magnification of the eyepiece of the microscope is The overall magnification of the microscope is the product of the individual magnifications

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 For example, you could not observe an atom (d  0.1 nm) with visible light (λ 500 nm)

Telescopes Two fundamental types of telescopes Refracting telescope uses a combination of lenses to form an image Reflecting telescope uses a curved mirror and a lens to form an image 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 distant object The image is near the focal point of the eyepiece The two lenses are separated by the distance ƒ o + ƒ e which corresponds to the length of the tube The eyepiece forms an enlarged, inverted image of the first image

Angular Magnification of a Telescope The angular magnification depends on the focal lengths of the objective and eyepiece Angular magnification is particularly important for observing nearby objects Very distant 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

Examples of Telescopes Reflecting Telescopes Largest in the world are 10 m diameter Keck telescopes on Mauna Kea in Hawaii Largest single mirror in US is 5 m diameter on Mount Palomar in California Refracting Telescopes Largest in the world is Yerkes Observatory in Wisconsin Has a 1 m diameter

Resolution The ability of an optical system to distinguish between closely spaced objects is limited due to the wave nature of light If two sources of light are close together, they can be treated as non- coherent sources Because of diffraction, the images consist of bright central regions flanked by weaker bright and dark rings

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 For the images to be resolved, the angle subtended by the two sources at the slit must be greater than θ min

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

Resolving Power of a Diffraction Grating If λ 1 and λ 2 are nearly equal wavelengths between which the grating spectrometer can just barely distinguish, the resolving power, R, of the grating is 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

Michelson Interferometer The Michelson Interferometer is an optical instrument that has great scientific importance 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 a partially silvered mirror M One ray is reflected to M 1 and the other transmitted to M 2 After reflecting, the rays combine to form an interference pattern The glass plate ensures 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 M 1 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