Chapter 8: Lights and Optics
What wave phenomenon do diffraction fringes result from?
Fringes result from constructive and destructive interference between light rays.
Multiple lens systems (focal length)
Lenses in contact are a series of lenses with negligible distances between them. These systems behave as a single lens with equivalent focal length given by 1/f = 1/f1 + 1/f2 + 1/f3 +...+ 1/fn
Magnification (m)
a dimensionless value that is the ratio of the image distance to the object distance: By extension, the magnification also gives the ratio of the size of the image to the size of the object. negative magnification signifies an inverted image, while a positive value signifies an upright image. If |m| < 1, the image is smaller than the object (reduced); if |m| > 1, the image is larger than the object (enlarged); and if |m| = 1, the image is the same size as the object.
Total internal reflection
a phenomenon in which all the light incident on a boundary is reflected back into the original material, results with any angle of incidence greater than the critical angle, θc, figure 8.10 At the incident angle of θc, the refracted angle is equal to 90°; at incident angles above 90°, total internal reflection occurs.
Reflection
rebounding of incident light waves at the boundary of a medium. Light waves that are reflected are not absorbed into the second medium; rather, they bounce off of the boundary and travel back through the first medium Normal: a line drawn perpendicular to the boundary of a medium; all angles in optics are measured from the normal, not the surface of the medium
Index of refraction (n)
the index of refraction of a vacuum is 1, by definition; for all other materials, the index of refraction will be greater than 1. For air, n is essentially equal to 1 because the speed of light in air is extremely close to c.
Law of reflection
theta1=theta2
Ray diagram
useful for getting an approximation of where an image is. On Test Day, ray diagrams can be helpful for a quick determination of the type of image that will be produced by an object some distance from the mirror (real vs. virtual, inverted vs. upright, and magnified vs. reduced). Ray diagrams should be used with caution, however: under the pressure of Test Day, it can be easy to draw them incorrectly. Therefore, it is important to practice drawing ray diagrams to avoid careless errors on Test Day, and it is also important to be familiar with how to solve optics questions mathematically. When drawing a ray diagram, there are three important rays to draw. For a concave mirror, a ray that strikes the mirror parallel to the axis (the normal passing through the center of the mirror) is reflected back through the focal point (green lines in Figure 8.6 and 8.7). A ray that passes through the focal point before reaching the mirror is reflected back parallel to the axis (red lines). A ray that strikes the mirror at the point of intersection with the axis is reflected back with the same angle measured from the normal (blue lines). In Figure 8.6a, the object is placed beyond F, and the image produced is real, inverted and magnified. In Figure 8.6b, the object is placed at F, and no image is formed because the reflected light rays are parallel to each other. In terms of the mirror equation, we say that the image distance i = ∞ here. For the scenario in Figure 8.6c, the object is placed between F and the mirror, and the image produced is virtual, upright, and magnified.
Visible Spectrum
wavelengths between(400 nm to 700 nm)
Speed of light from frequency and wavelength
where c is the speed of light in a vacuum and, to a first approximation, also in air, f is the frequency, and λ is the wavelength.
Positions of dark fringes (minima) on the screen (young's experiment)
where d is the distance between the two slits, θ is the angle between the line drawn from the midpoint between the two slits to the dark fringe and the normal, n is an integer indicating the number of the fringe, and λ is the wavelength of the incident wave. Note that bright fringes are halfway between dark fringes.
True or False: Incident angle is always measured with respect to the normal.
True. In optics, incident angles are always measured relative to the normal.
True or False: Maxima in diffraction patterns are always equidistant between two minima.
True. Maxima and minima alternate in a diffraction pattern. A maximum is equidistant between two minima, and a minimum is equidistant between two maxima.
What are the boundaries of the visible spectrum? How does the range of the visible spectrum compare to the range of the full electromagnetic spectrum?
Visible light ranges from wavelengths of about 400 nm to 700 nm. This is in comparison to the entire EM spectrum which ranges from wavelengths of nearly 0 to 109 m.
Electromagnetic spectrum preview
includes radio waves on one end (long wavelength, low frequency, low energy) and gamma rays on the other (short wavelength, high frequency, high energy). Between the two extremes, we find, in order from lowest energy to highest energy, microwaves, infrared, visible light, ultraviolet, and x-rays
Reflection in a plane mirror
all incident angles (theta) are equal to their respective reflected angles (theta')
Blackbody
The term blackbody refers to an ideal absorber of all wavelengths of light, which would appear completely black if it were at a lower temperature than its surroundings.
What are the two mathematical relationships between image distance and object distance?
1/f = 1/o + 1/i m= -I/o
Speed of light (c)
3 x 10^8 m/s
Ray diagrams for convex (diverging) mirrors
A single diverging mirror forms only a virtual, upright, and reduced image, regardless of the position of the object. The further away the object, the smaller the image will be. To quickly remember these rules, recall the convenience store security mirrors mentioned at the beginning of the chapte
Single Slit
Although it is usually safe to assume that nonrefracted light travels in a straight line, there are situations where light will not actually travel in a straight-line path. When light passes through a narrow opening (an opening with a size that is on the order of light wavelengths), the light waves seem to spread out (diffract), as is shown in Figure 8.15. As the slit is narrowed, the light spreads out more. Light emerges from a narrow slit in a wide arc, not a narrow beam.
Plane Mirrors (real or virtual images)
An image is said to be real if the light actually converges at the position of the image. An image is virtual if the light only appears to be coming from the position of the image but does not actually converge there. One of the distinguishing features of real images is the ability of the image to be projected onto a screen.
Critical angle (θc)
As the incident angle is increased, the refracted angle also increases, and eventually, a special incident angle called the critical angle (θc) is reached, for which the refracted angle θ2 equals 90 degrees. At the critical angle, the refracted light ray passes along the interface between the two media. The critical angle can be derived from Snell's law if θ2 = 90°, such that
Multiple lens systems (power)
Because power is the reciprocal of focal length, the equivalent power is A good example of lenses in contact is a corrective contact lens worn directly on the eye. In this case, the cornea of the eye (a converging lens) is in contact with a contact lens (either converging or diverging, depending on the necessary correction), and their powers would be added.
Chromatic aberration
Chromatic aberration, shown in Figure 8.14, is a dispersive effect within a spherical lens. Depending on the thickness and curvature of the lens, there may be significant splitting of white light, which results in a rainbow halo around images. This phenomenon is corrected for in visual lenses like eyeglasses and car windows with special coatings that have different dispersive qualities from the lens itself. ight dispersion within the glass lens leads to the formation of a rainbow halo at the edge of the image.
Circular Polarization
Circular polarization is a rarely seen natural phenomenon that results from the interaction of light with certain pigments or highly specialized filters. Circularly polarized light has a uniform amplitude but a continuously changing direction, which causes a helical orientation in the propagating wave, as shown in Figure 8.20. The helix has average electrical field vectors and magnetic field vectors that lie perpendicular to one another, like other waves, with maxima that fall on the outer border of the helix.
Diffraction gratings
Diffraction gratings consist of multiple slits arranged in patterns. Diffraction gratings can create colorful patterns similar to a prism as the different wavelengths interfere in characteristic patterns. For example, the organization of the grooves on a CD or DVD act like a diffraction grating, creating an iridescent rainbow pattern on the surface of the disc. Thin films may also cause interference patterns because light waves reflecting off the external surface of the film interfere with light waves reflecting off the internal surface of the film, as shown in Figure 8.18. Common examples of thin films are soap bubbles or oil puddles in wet parking lots. Note that the interference here is not between diffracted rays, but between reflected rays.
Diffraction
Diffraction refers to the spreading out of light as it passes through a narrow opening or around an obstacle. Interference between diffracted light rays lead to characteristic fringes in slit-lens and double-slit systems. Diffraction and interference are significant evidence for the wave theory of light.
How does the diffraction pattern for a single slit differ from a slit with a thin lens?
Diffraction through a single slit does not create characteristic fringes when projected on a screen, although the light does spread out. When a lens is introduced into the system, the additional refraction of light causes constructive and destructive interference, creating fringes.
Define the following terms:
Dispersion: Dispersion is the tendency for different wavelengths of light to experience different degrees of refraction in a medium, leading to separation of light into the visible spectrum (a rainbow). Aberration: Aberration (spherical or chromatic) is the alteration or distortion of an image as a result of an imperfection in the optical system.
electromagnetic waves
Electromagnetic waves are transverse waves because the oscillating electric and magnetic field vectors are perpendicular to the direction of propagation. The electric field and the magnetic field are also perpendicular to each other. The electric field (E) oscillates up and down the page; the magnetic field (B) oscillates into and out of the page. figure 8.1
True or False: Light waves are longitudinal because the direction of propagation is perpendicular to the direction of oscillation.
False. Light waves are transverse because the direction of propagation is perpendicular to the direction of oscillation.
Concave surface
For a concave surface, the center of curvature and the radius of curvature are located in front of the mirror. If we were to look from the inside of a sphere to its surface, we would see a concave surface. converging mirrors; converge after they reflect
Convex surface
For a convex surface, the center of curvature and the radius of curvature are behind the mirror. On the other hand, if we were to look from outside the sphere, we would see a convex surface. diverging mirrors; light rays diverge after they reflect
Sign conventions for lenses
For both lenses and mirrors, positive magnification represents upright images, and negative magnification means inverted images. Also, for both lenses and mirrors, a positive image distance means that the image is real and is located on the real (R) side, whereas a negative image distance means that the image is virtual and located on the virtual (V) side.
Multiple lens systems (magnification)
For lenses not in contact, the image of one lens becomes the object of another lens. The image from the last lens is considered the image of the system. Microscopes and telescopes are good examples of these systems. The magnification for the system is
Real lenses/Lensmaker equation
For lenses where the thickness cannot be neglected, the focal length is related to the curvature of the lens surfaces and the index of refraction of the lens by the lensmaker's equation: where n is the index of refraction of the lens material, r1 is the radius of curvature of the first lens surface and r2 is the radius of curvature of the second lens surface. The eye is a complex refractive instrument that uses real lenses. The cornea acts as the primary source of refractive power because the change in refractive index from air is so significant. Then, light is passed through an adaptive lens that can change its focal length before reaching the vitreous humor. It is further diffused through layers of retinal tissue to reach the rods and cones. At this point, the image has been focused and minimized significantly, but is still relatively blurry. Our nervous system processes the remaining errors to provide a crisp view of the world.
Slit-Lens System
If a lens is placed between a narrow slit and a screen, a pattern is observed consisting of a bright central fringe with alternating dark and bright fringes on each side, as shown in Figure 8.16. The central bright fringe (maximum) is twice as wide as the bright fringes on the sides, and as the slit becomes narrower, the central maximum becomes wider. The location of the dark fringes (minima) is given by the formula where a is the width of the slit, θ is the angle between the line drawn from the center of the lens to the dark fringe and the axis of the lens, n is an integer indicating the number of the fringe, and λ is the wavelength of the incident wave. Note that bright fringes are halfway between dark fringes.
Image distance
If the image has a positive distance (i > 0), it is a real image, which implies that the image is in front of the mirror. If the image has a negative distance (i < 0), it is virtual and thus located behind the mirror. Use 1/f = 1/o + 1/I = 2/r
Thomas Young, double slit experiment
In his famous double-slit experiment, Thomas Young showed that the diffracted rays of light emerging from two parallel slits can interfere with one another. This was a landmark finding that contributed to understanding of light as a wave. Figure 8.17 shows the typical setup for Young's double-slit experiment. When monochromatic light (light of only one wavelength) passes through the slits, an interference pattern is observed on a screen placed behind the slits. Regions of constructive interference between the two light waves appear as bright fringes (maxima) on the screen. Conversely, in regions where the light waves interfere destructively, dark fringes (minima) appear.
White
Light that contains all the colors in equal intensity is perceived as white.
Describe the bending of light when moving from a medium with low refractive index to high refractive index and from a medium with high refractive index to low refractive index:
Low n to high n: Light will bend toward the normal when going from a medium with low n to high n. High n to low n: Light will bend away from the normal when going from a medium with high n to low n if the incident angle is larger than the critical angle (θc), total internal reflection will occur.
Sign convention for a single mirror
Note that on the MCAT, for almost all problems involving mirrors, the object will be placed in front of the mirror. Thus, the object distance o is almost always positive.
Thin spherical lenses
On the MCAT, lenses generally have negligible thickness. Because light can travel from either side of a lens, a lens has two focal points, with one on each side. The focal length can be measured in either direction from the center. For thin spherical lenses, the focal lengths are equal, so we speak of just one focal length for the lens as a whole. Figure 8.11a illustrates that a converging lens is always thicker at the center, while Figure 8.11b illustrates that a diverging lens is always thinner at the center. The basic formulas for finding image distance and magnification for spherical mirrors also apply to lenses. The object distance o, image distance i, focal length f, and magnification m, are related by the equations 1/f = 1/o + 1/I = 2/r m=-i/o
Power (P)
Optometrists often describe a lens in terms of its power (P). This is measured in diopters, where f (the focal length) is in meters and is given by the equation P=1/f P has the same sign as f and is, therefore, positive for a converging lens and negative for a diverging lens. People who are nearsighted (can see near objects clearly) need diverging lenses, while people who are farsighted (can see distant objects clearly) need converging lenses. Bifocal lenses are corrective lenses that have two distinct regions—one that causes convergence of light to correct for farsightedness (hyperopia) and a second that causes divergence of light to correct for nearsightedness (myopia) in the same lens.
Plane mirrors
Parallel incident light rays remain parallel aer reflection from a plane mirror; that is, plane mirrors—being flat reflective surfaces—cause neither convergence nor divergence of reflected light rays. Because the light does not converge at all, plane mirrors always create virtual images. In a plane mirror, the image appears to be the same distance behind the mirror as the object is in front of it In other words, plane mirrors create the appearance of light rays originating behind the mirrored surface. Because the reflected light remains in front of the mirror but the image appears behind the mirror, the image is virtual. Plane mirrors include most of the common mirrors found in our homes. To assist in our discussion of spherical mirrors, plane mirrors can be conceptualized as spherical mirrors with an infinite radius of curvature.
Plane mirrors and image distance
Plane mirrors can be thought of as spherical mirrors with infinitely large focal distances. As such, for a plane mirror, r = f = ∞, and the equation becomes or 1/o + 1/I=0 or i = -o. This can be interpreted as saying the virtual image is at a distance behind the mirror equal to the distance the object is in front of the mirror.
How does the application of a polarized filter impact the wavelength of light passing through the filter?
Plane polarization has no effect on the wavelength (or frequency or speed) of light. Polarization does affect the amount of light passing through a medium and light intensity.
plane-polarized (linearly polarized) light
Plane-polarized (or linearly polarized) light is light in which the electric fields of all the waves are oriented in the same direction (that is, their electric field vectors are parallel). It follows that their magnetic fields vectors are also parallel, but convention dictates that the plane of the electric field identifies the plane of polarization. Unpolarized light has a random orientation of its electric field vectors; sunlight and light emitted from a light bulb are prime examples.
Contrast plane-polarized and circularly polarized light:
Plane-polarized light contains light waves with parallel electric field vectors. Circularly polarized light selects for a given amplitude and has a continuously rotating electric field direction.
Snell's Law
Refracted rays of light obey Snell's law as they pass from one medium to another: where n1 and θ1 refer to the medium from which the light is coming and n2 and θ2 refer to the medium into which the light is entering. Note that θ is once again measured with respect to the normal From Snell's law, we can see that when light enters a medium with a higher index of refraction (n2 > n1), it bends toward the normal (sin θ2 < sin θ1; therefore, θ2 < θ1), as shown in Figure 8.9. Conversely, if the light travels into a medium where the index of refraction is smaller (n2 < n1), the light will bend away from the normal (sin θ2 > sin θ1; therefore, θ2 > θ1).
Refraction
Refraction is the bending of light as it passes from one medium to another and changes speed. The speed of light through any medium is always less than its speed through a vacuum. Remember that the speed of light in a vacuum, c, is equal to 3x10^8 m/s The speed of light in air is just slightly lower that this value. but just use same number
Spherical mirrors
Spherical mirrors come in two varieties: concave and convex. The word spherical implies that the mirror can be considered a spherical cap or dome taken from a much larger spherically shaped mirror. Spherical mirrors have an associated center of curvature (C) and a radius of curvature (r)
Geometrical optics
The behavior of light at the boundary of a medium or interface between two media is described by the theory of geometrical optics. Geometrical optics explains reflection and refraction, as well as the applications of mirrors and lenses.
Center of curvature (C) and Radius of curvature (r)
The center of curvature is a point on the optical axis located at a distance equal to the radius of curvature from the vertex of the mirror; in other words, the center of curvature would be the center of the spherically shaped mirror if it were a complete sphere. the radius of curvature (r) is the distance between C and the mirror. (diagram: concave mirror)
Color
The color of an object that does not emit its own light is dependent on the color of light that it reflects. Thus, an object that appears red is one that absorbs all colors of light except red. This implies that a red object under green illumination will appear black because it absorbs the green light and has no light to reflect.
Real vs. Virtual
The designations of real and virtual are oen a point of confusion for students because they are on opposite sides when comparing mirrors and lenses. To identify the real side (R), remember that the real side is where light actually goes aer interacting with the lens or mirror. For mirrors, light is reflected and, therefore, stays in front of the mirror. Hence, for a mirror, the real side is in front of the mirror, and the virtual side is behind the mirror. For lenses, the convention is different: because light travels through the lens and comes out on the other side, the real side is on the opposite side of the lens from the original light source, and the virtual side is on the same side of the lens as the original light source. Although the object of a single lens is on the virtual side, this does not make the object virtual. Objects are real, with a positive object distance, unless they are placed in certain multiple lens systems in which the image of one lens becomes the object for another (a scenario which is very rarely encountered on the MCAT).
Electromagnetic spectrum
The electromagnetic spectrum describes the full range of frequencies and wavelengths of electromagnetic waves. Wavelengths are oen given in the following units: mm (10-3 m), μm (10-6 m), nm (10-9 m), and Å (ångström, 10-10 m). The full spectrum is broken up into many regions, which in descending order of wavelength are radio (109-1 m), microwave (1 m-1 mm), infrared (1 mm-700 nm), visible light (700-400 nm), ultraviolet (400-50 nm), x-ray (50-10-2 nm), and γ-rays (less than 10-2 nm)
Key variables in geometrical optics
The focal length (f) is the distance between the focal point (F) and the mirror. Note that for all spherical mirrors, f=r/2 where the radius of curvature (r) is the distance between C and the mirror. The distance between the object and the mirror is o; the distance between the image and the mirror is i. There is a simple relationship between these four distances: 1/f = 1/o + 1/I = 2/r
How does double-slit diffraction and interference differ from single-slit diffraction?
The image formed during double-slit diffraction contains fringes because light rays constructively and destructively interfere. A single slit forms an image of a wide band of light, spread out from its original beam.
Visible region
The only part of the spectrum that is perceived as light by the human eye is the visible region. Within this region, different wavelengths are perceived as different colors, with violet at one end of the visible spectrum (400 nm) and red at the other (700 nm).
Specific rotation
The optical activity of a compound, due to the presence of chiral centers, causes plane-polarized light to rotate clockwise or counterclockwise by a given number of degrees relative to its concentration (its specific rotation). Remember that enantiomers, as nonsuperimposable mirror images, will have opposite specific rotations.
Polarizers
There are filters called polarizers, oen used in cameras and sunglasses, which allow only light with an electric field pointing in a particular direction to pass through. If one passes a beam of light through a polarizer, it will only let through that portion of the light parallel to the axis of the polarizer. If a second polarizer is then held up to the first, the angle between the polarizers' axes will determine how much light passes through. When the polarizers are aligned, all the light that passes through the first polarizer also passes through the second. When the second polarizer is turned so that its axis is perpendicular, no light gets through at all.
Lenses
There is an important difference between lenses and mirrors, aside from the fact that lenses refract light while mirrors reflect it. When working with lenses, there are two surfaces that affect the light path. For example, a person wearing glasses sees light that travels from an object through the air into the glass lens (first surface). Then the light travels through the glass until it reaches the other side, where again it travels out of the glass and into the air (second surface). The light is refracted twice as it passes from air to lens and from lens back to air.
Index of refraction equation (snells law pictured)
When light is in any medium besides a vacuum, its speed is less than c. For a given medium n= c/v where c is the speed of light in a vacuum, ν is the speed of light in the medium, and n is a dimensionless quantity called the index of refraction of the medium
Rectilinear propagation
When light travels through a homogeneous medium, it travels in a straight line. This is known as rectilinear propagation.
Dispersion
When various wavelengths of light separate from each other, this is called dispersion. The most common example of dispersion is the splitting of white light into its component colors using a prism. If a source of white light is incident on one of the faces of a prism, the light emerging from the prism is spread out into a fan-shaped beam, as shown in Figure 8.13. This occurs because violet light has a smaller wavelength than red light and so is bent to a greater extent. Because red experiences the least amount of refraction, it is always on top of the spectrum; violet, having experienced the greatest amount of refraction, is always on the bottom of the spectrum. Note that as light enters a medium with a different index of refraction, the wavelength changes but the frequency of the light does not.
Interference
When waves interact with each other, the displacements of the waves add together in a process called interference
X Ray diffraction
X-ray diffraction uses the bending of light rays to create a model of molecules. X-ray diffraction is oen combined with protein crystallography during protein analysis. Dark and light fringes do not take on a linear appearance, but rather a complex two dimensional image
Spherical aberration
spherical aberration is a blurring of the periphery of an image as a result of inadequate reflection of parallel beams at the edge of a mirror or inadequate refraction of parallel beams at the edge of a lens. This creates an area of multiple images with very slightly different image distances at the edge of the image, which appears blurry. Parallel rays are not perfectly reflected or refracted through the focal point, leading to blurriness at the periphery of the image.
Order the types of electromagnetic radiation from highest energy to lowest energy. What other property of light follows the same trend?
γ-rays > x-rays > ultraviolet > visible light > infrared > microwaves > radio. Frequency follows the same trend as energy, whereas wavelength follows the opposite trend.
