8. Light and Optics

The full electromagnetic spectrum 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.

When drawing a ray diagram, there 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.

A ray that passes through the focal point before reaching the mirror is reflected back parallel to the axis. 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. 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.

KEY CONCEPT 2

Any time an object is at the focal point of a converging mirror, the reflected rays will be parallel, and thus, the image will be at infinity.

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

Because power is the reciprocal of focal length, the equivalent power is P=P1+P2+P3+Pn A good example of lenses in contact is a corrective contact lens worn directly on the eye. In this case, the lens 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.

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. 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.

KEY CONCEPT 1

Concave mirrors are converging mirrors. Convex mirrors are diverging mirrors. The reverse is true for lenses.

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 ).

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 ).

X-ray diffraction uses the bending of light rays to create a model of molecules. X-ray diffraction is often 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.

A changing magnetic field can cause a change in an electric field, and a changing electric field can cause a change in a magnetic field. Because of the reciprocating nature of these two fields, we can see how electromagnetic waves occur in nature. Each oscillating field causes oscillations in the other field completely independent of matter, so electromagnetic waves can even travel through a vacuum.

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 electromagnetic spectrum describes the full range of frequencies and wavelengths of electromagnetic waves.

Electromagnetic waves vary in frequency and wavelength, but in a vacuum, all electromagnetic waves travel at the same speed, called the speed of light. This constant is represented by c. approximately To a first approximation—and for the purposes of all MCAT-related equations—electromagnetic waves also travel in air with this speed. In reference to electromagnetic waves, the familiar equation ν = f λ becomes c = f λ 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.

The designations of real and virtual are often 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 after 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).

Focal lengths and radii of curvature have a simpler sign convention. For both mirrors and lenses, converging species have positive focal lengths and radii of curvature, and diverging species have negative focal lengths and radii of curvature. Remember that lenses have two focal lengths and two radii of curvature because they have two surfaces. For a thin lens where thickness is negligible, the sign of the focal length and radius of curvature are generally given based on the first surface the light passes through.

There are filters called polarizers, often 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.

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). 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.

If we were to look from the inside of a sphere to its surface, we would see a concave surface. On the other hand, if we were to look from outside the sphere, we would see a convex surface. For a concave surface, the center of curvature and the radius of curvature are located in front of the mirror. For a convex surface, the center of curvature and the radius of curvature are behind the mirror. Concave mirrors are called converging mirrors and convex mirrors are called diverging mirrors because they cause parallel incident light rays to converge and diverge after they reflect, respectively

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.

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.

KEY CONCEPT 8

It is important to realize that concave mirrors and convex lenses are both converging and thus have similar properties. Convex mirrors and concave lenses are both diverging and also have similar properties.

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.

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).

Light that contains all the colors in equal intensity is perceived as white. 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. 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.

Reflection is the 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

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 m = m1 × m2 × m3 × ··· × mn

Sign Conventions for Lenses

Note that the sign conventions change slightly 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.

Plane-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.

One of the most common applications of planepolarized light on the MCAT is in the classification of stereoisomers. 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.

In general, images created by a mirror can be either real or virtual. 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.

Parallel incident light rays remain parallel after 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.

A ray diagram is 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.

KEY CONCEPT 5

Remember that when light enters a medium with a higher index of refraction, it bends toward the normal. When light enters a medium with a lower index of refraction, it bends away from the normal.

Spherical mirrors and lenses are imperfect. They are therefore subject to specific types of errors or aberrations.

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.

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.

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.

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 and -m/o

When light travels through a homogeneous medium, it travels in a straight line. This is known as rectilinear propagation.

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.

KEY CONCEPT 9

The electric fields of unpolarized light waves exist in all three dimensions: the direction of the wave's propagation is surrounded by electric fields in every plane perpendicular to that direction. Polarizing light limits the electric field's oscillation to only two dimensions.

KEY CONCEPT 4

The focal length of converging mirrors (and converging lenses) will always be positive. The focal length of diverging mirrors (and diverging lenses) will always be negative.

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.

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.

On the MCAT, you will most often use this equation to calculate the image distance for all types of mirrors and lenses. 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. 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 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.

The magnification (m) is a dimensionless value that is the ratio of the image distance to the object distance: m= -i/o

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.

There are several important lengths associated with mirrors. 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 three distances: 1/f=1/o+1/i=2/r While it is not important which units of distance are used in this equation, it is important that all values used have the same units as each other.

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.

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. 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.

Chromatic aberration, 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 than the lens itself.

KEY CONCEPT 7

To find where the image is (for a lens), draw the following rays and find a point where any two intersect. This point of intersection marks the tip of the image. If the rays you draw do not appear to intersect, extend them to the same side of the lens from which the light came, creating a virtual image.

KEY CONCEPT 3

To find where the image is (for a mirror), draw the following rays and find a point where any two intersect. This point of intersection marks the tip of the image. If the rays you draw do not appear to intersect, extend them to the other side of the mirror, creating a virtual image. Ray parallel to axis → reflects back through focal point Ray through focal point → reflects back parallel to axis Ray to center of mirror → reflects back at same angle relative to normal

KEY CONCEPT 6

Total internal reflection occurs as the light moves from a medium with a higher refractive index to a medium with a lower one.

When waves interact with each other, the displacements of the waves add together in a process called interference. In his famous doubleslit 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.

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.

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 the slit is narrowed, the light spreads out more.

As discussed earlier, the speed of light in a vacuum is the same for all wavelengths. However, when light travels through a medium, different wavelengths travel at different speeds. This fact implies that the index of refraction of a medium affects the wavelength of light passing through the medium because the index of refraction is related to the speed of the wave by n=c/v.

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.

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.

By extension, the magnification also gives the ratio of the size of the image to the size of the object. The orientation of the image (upright or inverted) can be determined: a 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.

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 a sin θ = n λ

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.

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. 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.

The positions of dark fringes (minima) on the screen can be found from the equation d sin θ=(n+1/2)λ

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.

Refracted rays of light obey Snell's law as they pass from one medium to another:

where n and θ refer to the medium from which the light is coming and n and θ refer to the medium into which the light is entering. Note that θ is once again measured with respect to the normal.

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 law of reflection is θ1 = θ2

where θ is the incident angle and θ is the reflected angle, both measured from the normal. The normal is 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.

When light travels from a medium with a higher index of refraction (such as water) to a medium with a lower index of refraction (such as air), the refracted angle is larger than the incident angle ( θ2 > θ1 ); that is, the refracted light ray bends away from the normal. 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 θ = 90°, such that

θc=sin(n2/n1)