Topic Summaries: Waves & Optics
The Physics Hypertextbook™
© 1998-2008 by Glenn Elert -- A Work in Progress
All Rights Reserved -- Fair Use Encouraged
prev | up | next
- Introduction
- Basic Properties of Waves
- A wave is a disturbance that propagates through a medium.
- Propagation describes the spreading of a disturbance
- Waves transfer energy, momentum, and information, but not mass.
- Classifying Waves by Medium
- Mechanical Waves: matter is the medium
- Sound is a mechanical wave
- Electromagnetic Waves: electric and magnetic fields are the media
- Light is an electromagnetic wave
- Gravitational Waves: the gravitational field is the medium.
(The existence of gravitational waves has not yet been confirmed.)
- Classifying Waves by Orientation
- Transverse Waves: disturbance is perpendicular to the direction of propagation
- All electromagnetic waves are transverse. This includes light.
- Crest: a point of maximal displacement in the positive direction
- Trough: a point of maximal displacement in the negative direction
- Longitudinal Waves: disturbance is parallel to the direction of propagation
- Sound is a longitudinal wave
- Compression or Condensation: a region where the medium is under compression
- Rarefaction or Dilation: a region where the medium is under tension
- Surface Waves or Complex Waves: a combination transverse-longitudinal wave, forms near the surface of some media
- Torsional Waves: disturbance causes the medium to twist
- Classifying Waves by Duration
- adj. episodic; noun pulse: disturbance is momentary and sudden
- adj. periodic, harmonic; noun wave train: disturbance repeats at regular intervals
- Classifying Waves by Appearance
- Traveling Waves: appear to move
- Standing Waves: do not appear to move
- Waves propagate with a finite speed (sometimes called the wave speed) that depends upon …
- the type of wave,
- the composition of the medium, and
- the state of the medium
- Wave Phenomena
- Periodic Waves
- Fundamentals
- A wave is a disturbance that propagates.
- A cycle is a sequence of events that repeats.
- A cycle is periodic if it repeats with a characteristic time.
- Characteristics
- Amplitude (A) is the maximum absolute value of a periodically varying quantity.
- Amplitude has the unit of the quantity that is changing (ex. displacement, pressure, field strength, etc.)
- Period (T) is the time between successive cycles of a repeating sequence of events.
- T = t/n (time per number of cycles)
- The SI unit of period is the second [s].
- Frequency (ƒ) is the number of cycles of a repeating sequence of events in a unit interval of time.
- ƒ = n/t (number of cycles per time)
- Frequency and period are reciprocals (or inverses) of one another: ƒ = 1/T.
- The SI unit of frequency is the hertz [Hz = 1/s = s−1].
- Phase (ϕ) is the stage of development of a periodic process.
- Two points on a wave with the same phase have the same …
- quantity of disturbance (ex. displacement) and
- rate of change of disturbance (ex. velocity).
- Phase is an angular quantity.
- Adjacent points in phase are separated by one complete cycle.
- Adjacent points out of phase are separated by half a cycle.
- The SI unit of phase is the radian, which is itself a unitless ratio [rad = m/m = Pa/Pa = (V/m)/(V/m) = etc.].
- Wavelength (λ)
is the distance between any point on a periodic wave and the next point corresponding to the same portion of the wave measured along the path of propagation.
- Wavelength is measured between adjacent points in phase.
- The SI unit of wavelength is the meter [m].
- Speed (v) is …
- v = Δs/Δt the rate of change of distance with time by definition and
- v = λƒ the product of wavelength and frequency for periodic waves.
- Frequency and wavelength are inversely proportional.
- Lower frequency waves have longer wavelengths.
- Higher frequency waves have shorter wavelengths.
- The speed of a wave is sometimes known as its wave speed
- The SI unit of speed is the meter per second [m/s].
- One-Dimensional Wave Equation
- ƒ (x, t) = A sin (2π (ft − x/λ) + ϕ)
- A, amplitude
- ƒ, frequency
- λ, wavelength
- ϕ, phase
- ƒ (x, t) = A sin (ωt − kx + ϕ)
- A, amplitude
- ω, angular frequency, ω = 2πf
- k, wave number k = 2π/λ
- ϕ, phase
- Interference & Superposition
- Objects made of matter
- are material, tangible, corporeal, and definite
- cannot occupy the same place at the same time
- exchange energy and momentum during collisions
- Waves
- are immaterial, intangible, incorporeal, and indefinite
- can occupy the same place at the same time
- pass through each other without effect
- Principle of Linear Superposition
- When waves occupy the same place at the same time they interfere
or superpose
- The resultant disturbance is the sum of the individual disturbances
at every point in space and time
- Reflection, Transmission, Absorption
- Boundary & Interface
- A boundary is the end, edge, or face of
a finite medium
- A "rigid" boundary is known as a fixed
end or closed end.
- A "loose" boundary is known as a free
end or open end.
- An interface is a boundary shared by two
media.
- To a wave entering a medium with a slower wave speed,the interface
is more like a fixed end than a free end.
- To a wave entering a medium with a faster wave speed, the interface
is more like a free end than a fixed end.
- Reflection, Transmission, Absorption
- When a wave is incident on a boundary or
interface it is partially reflected, partially
transmitted, and partially absorbed.
- Energy and momentum are conserved in the process
- An echo is a reflected wave, especially
a reflected sound wave
- Properties of Materials
- The degree of reflection, transmission, and absorption depends upon
the two media and the frequency of the incident wave.
- A material is …
- opaque if it prevents the transmission
of a wave,
- transparent if it allows the transmission
of a wave, or
- absorbent if it prevents both the reflection
and the transmission of a wave.
- Wave Characteristics at an Interface
- Amplitude is related to energy.
- Ereflected + Etransmitted = Eincident,
conservation of energy, no absorption
- Ereflected + Etransmitted < Eincident,
work-energy theorem, absorption present
- Speed is affected by medium.
- The reflected wave speed is the same as the incident wave speed.
- The transmitted wave speed may be different from incident wave speed.
- Frequency is determined at the source.
- Frequency and period remain constant upon transmission and reflection.
- Wavelength is affected by speed and frequency.
- Wavelength is directly proportional to speed.
- Wavelength is inversely proportional to frequency.
- Phase …
- remains constant on transmission, but
- may be affected by reflection.
- Reflection from a free end leaves phase unchanged. (The wave
is reflected in phase.)
- Reflection from a fixed end inverts the wave. (The wave is reflected
out of phase.)
- Standing Waves
- Traveling waves …
- Standing waves …
- do not appear to propagate
- are produced by the interference of two waves traveling in opposite
directions with the same frequency and amplitude
- Positions on a standing wave:
- node: a point where the amplitude is
zero or a minimum
(always form at fixed ends)
- antinode: a point where the amplitude
is a maximum
(always form at free ends)
- Resonance
- Resonance is the dramatic increase in amplitude of a periodic system
that occurs when the driving frequency
applied equals the natural frequency
of the system
- Standing waves form during resonance (but resonance does not always
lead to the formation of standing waves)
- A wave moving in a medium of finite length, can interfere with its own
reflection to produce a standing wave if it has the same frequency as
one of the natural frequencies of the medium
- Harmonics
- are the set of all possible standing waves in a system
- are countably infinite in number (form a countable infinite set in the
manner of whole numbers)
- Groups of harmonics:
- fundamental: harmonic with the lowest
frequency and longest wavelength
- overtones: harmonics other than the
fundamental
- When standing waves form in a linear medium that has …
- two fixed ends or two free ends …
- a whole number of half wavelengths fit inside the medium and
- the overtones are whole number multiples of the fundamental frequency.
- one fixed end and one free end …
- an odd number of quarter wavelengths fit inside the medium and
- the overtones are odd multiples of the fundamental frequency.
- Higher-dimensional cases
- You probably don't need to worry about it.
- Diffraction
- Interference in Two and Three Dimensions
- Sound
- The Nature of Sound
- Sound is a mechanical, longitudinal wave.
- As a mechanical wave, sound requires a medium.
- Sound cannot propagate through a vacuum.
- There is no sound in outer space.
- As a longitudinal wave, sound is a rapid variation in pressure that propagates.
- Regions of above normal pressure (regions under compression) are called compressions or condensations.
- Regions of below normal pressure (regions under tension) are called rarefactions or dilations.
- Sound is produced by small and rapid pressure changes.
- Vibrating objects produce periodic sound waves.
- Implosive or explosive pressure changes produces sound pulses.
- Vortex shedding is another source of periodic sound waves.
- The speed of sound depends upon the medium and its state.
- Sound usually travels fast in gases, faster in liquids, and fastest in solids.
- The speed of sound in air increases with temperature.
- There are several formulas for calculating the speed of sound in air as a function of temperature.
- The speed of sound in a medium is largely independent of amplitude and frequency.
- The amplitude of a sound wave corresponds to its intensity or loudness.
- The intensity of a sound is .
- a measure of its power density
- usually measured on a logarithmic scale.
- discussed in more detail in another section of this book
- The loudness of a sound is its intensity as perceived by the human ear.
- The volume knob on a television, radio, etc. should really be given a different name.
- The frequency of a sound wave corresponds to its pitch.
- The upper frequency limit for human hearing is around 18,000 to 20,000 Hz.
- Frequencies above the range of human hearing are ultrasonic.
- The lower frequency limit for human hearing is around 18 to 20 Hz.
- Frequencies below the range of human hearing are infrasonic.
- The frequency of a sound wave does not change as the sound wave propagates.
- Wavelength is inversely proportional to frequency (λ ∝ 1 / ƒ).
- Large objects generally produce long wavelength, low frequency sounds.
- Small objects generally produce short wavelength, high frequency sounds.
- The ability of an animal or electronic sensor to identify the location or direction of origin of a sound is known as sound localization.
- Sound localization requires two or more …
- sense organs (typically ears or antennae) or
- electromechanical detectors (typically microphones)
devoted to hearing …
- in different locations (left and right sides of the head, for example) or
- with different orientations (facing to the left or to the right).
- All methods of sound localization rely on the difference in some characteristic as perceived or measured by the two organs or detectors.
- interaural level difference -- loudness, intensity, or amplitude
- interaural time differences -- time of arrival
- interaural phase difference -- phase differences
- A reflected sound wave is known as an echo.
- Echoes can be used to determine the distance to a reflecting surface.
Where …
| s |
is the distance from the observer to the reflecting surface (note that this value is doubled since the sound has to go out and come back), |
| vsound |
is the speed of sound in the intervening medium, and |
| Δt |
is the time between when the pulse was transmitted and when the echo was received. |
- This method has applications in …
- animal echolocation
- sonar (an acronym for sound navigation and ranging)
- medical ultrasonography (the images generated are called sonograms).
- Intensity
- Amplitude, intensity, and loudness are often used interchangeably,
but the three terms have different meanings.
- Amplitude is a measure of the maximal
change in whatever quantity is varying in a wave.
- For sound waves, the varying quantity [and its SI unit] could be …
- position [m],
- density [kg/m3], or
- pressure [Pa].
- Position and density variations caused by sound waves are difficult
to measure directly since …
- their magnitudes are small and
- the period of sound waves are brief.
- Pressure variations caused by sound waves can be detected …
- by animals with their ears and
- by machines with microphones
- Intensity is an objective measure
of the mean power density of the energy transferred by a wave.
- The SI unit of intensity is the watt per
square meter.
- As an equation, intensity is defined as …
where …
| I |
intensity of the wave |
| 〈P〉 |
time-averaged transmitted power |
| A |
area through which the wave or
a portion of the wave is propagating. |
- When amplitude is measured by maximum displacement, intensity
is equal to …
where …
| I |
intensity of the wave |
| ρ |
density of the medium |
| f |
frequency of the wave |
| v |
speed of the wave |
| Δx2max |
maximum displacement of the particles
in the medium |
- When amplitude is measured by maximum gauge pressure, intensity
is equal to …
where …
| I |
intensity of the wave |
| P |
maximum gauge pressure |
| ρ |
density of the medium |
| v |
speed of the wave |
- Beats
- Music & Noise
- Döppler Effect (Sound)
- Shock Waves
- Physical Optics
- The Nature of Light
- Light is a transverse, electromagnetic wave that can be seen by humans.
- The wave nature of light was first illustrated through experiments on diffraction and interference.
- Like all electromagnetic waves, light can travel through a vacuum.
- The transverse nature of light can be demonstrated through polarization.
- Light is sometimes also known as visible light to contrast it from "ultraviolet light" and "infrared light".
- Other forms of electromagnetic radiation that are not visible to humans are sometimes also known informally as "light"
- Light is produced by one of two methods.
- Incandescence is the emission of light from "hot" matter (T ≳ 800 K).
- Luminescence is the emission of light when bound electrons fall to lower energy levels.
- The speed of light depends upon the medium through which it travels.
- The speed of light in a vacuum is a universal constant in all reference frames.
- All electromagnetic waves propagate at the speed of light in a vacuum.
- The speed of light in a medium is always slower the speed of light in a vacuum.
(The difference is usually negligible when the medium is air.)
- The speed of anything with mass is always less than the speed of light in a vacuum.
(The speed of light in a vacuum is the universal speed limit.)
- The speed of light in a vacuum is fixed at 299,792,458 m/s by the current definition of the meter.
- The amplitude of a light wave is related to its intensity.
- Intensity is the absolute measure of a light wave's power density.
- Brightness is the relative intensity as perceived by the average human eye.
- The frequency of a light wave is related to its color.
- Color is such a complex topic that it has its own section in this book.
- Monochromatic light can be described by only one frequency.
- Laser light is very nearly monochromatic.
- There are six simple, named colors in English (and many other languages) each associated with a band of monochromatic light. In order of increasing frequency they are red, orange, yellow, green, blue, and violet.
- Polychromatic light is compused of multiple frequencies.
- Every light source is essentially polychromatic.
- White light is very polychromatic.
- A graph of relative intensity vs. frequency is called a spectrum (plural: spectra).
Although frequently associated with light, the term can be applied to many phenomena.
- A continuous spectrum is one in which every frequency is present within some range.
- Blackbody radiators emit a continuous spectrum.
- A discrete spectrum is one in
which only a set of well defined and isolated frequencies are
present.
(A discrete spectrum is a finite collection of monochromatic light waves.)
- The excited electrons in a gas emit a discrete spectrum.
| |
|
|
| condition |
description |
spectrum |
| hotter than red hot |
incandescent |
continuous |
| excited electrons |
luminous |
discrete |
| |
|
|
- The wavelength of a light wave is inversely proportional to its frequency.
- Light is often described by it's wavelength in a vacuum.
- Light ranges in wavelength from 400 nm on the violet end to 700 nm on the red end of the visible spectrum.
- Wavelengths slightly shorter than 400 nm are said to be ultraviolet.
(They are "beyond violet" in terms of frequency.)
- Wavelengths slightly longer than 700 nm are said to be infrared.
(They are "below red" in terms of frequency.)
- Phase differences between light waves can produce visible interference effects.
(There are several sections in this book on interference phenomena and light.)
- Color
- Color is the perceptual quality of light.
(Color is a subjective response by the brain to light stimulating the retina.)
- Two visual regions have the same color if a difference between them cannot be perceived by the average human eye.
- The human eye can distinguish nearly ten million colors.
- Color as a visual response should not be confused with the "color" of a pigment, which is the color one would see when viewing that pigment under typical lighting conditions.
- Although they may not satisfy the definition of a color in some fashion sense; black, white, and gray each satisfy the physical and perceptual definitions of a color.
- The color of the light coming from an object has its origin in one or more of the following processes …
- emission: the object itself is a source of light with a color determined by its spectra
- reflection: certain frequencies are reflected from the object while others are not
- transmission: certain frequencies are transmitted through the object while others are not
- interference: certain frequencies are amplified by constructive interference while others are attenuated by destructive interference
- dispersion: the angular separation of a polychromatic light wave by frequency during refraction
- scattering: the preferential reradiation of certain frequencies of light striking small, dispersed particles
- There are six simple, named colors in English (and many other languages) each associated with a band of monochromatic light. In order of increasing frequency they are red, orange, yellow, green, blue, and violet.
- The range of frequencies corresponding to each band is subject to individual, cultural, and historical factors.
- Indigo is not included in this list as it is purely a historic artifact. The word is rarely used by contemporary speakers of English to describe a color.
- The sensation of purple cannot be produced using light of a single frequency, but only by combining light from the red and blue/violet bands (light from the extreme ends of the visible spectrum).
- At relatively low intensities …
- monochromatic light in the red, orange, and yellow bands appear brown.
- monochromatic light in the blue band is difficult to distinguish from violet.
- Humans perceive polychromatic mixtures of light as a single color,
which may or may not look like light from a monochromatic light source.
- Polychromatic light is described physically by a spectral power distribution (often just called a spectrum), which is a graph of intensity vs. wavelength (or frequency)
- Regions with different spectra that appear to be the same color are called metamers and the effect is called metamerism.
- Regions that are metamers when illuminated by one light source may not appear to be the same color when illuminated by another light source.
- The wide variety of colors visible to humans can be approximated by mixing only a small subset of colored light sources or colored pigments.
- Color mixing can be accomplished by …
- superposition (e.g., lamp overlap, filter overlap)
- rapid alternation, faster than the persistence of vision (e.g., biased LED, rotating color wheel)
- small, nearby elements (e.g., dithering, pixels, halftone dots, photo grain, pigment mixing)
- White light is a mixture of visible frequencies of electromagnetic radiation whose appearance approximates that of a blackbody radiator with its peak wavelength in the middle of the visible spectrum.
- There is no one frequency distribution that can be identified as white light. Human vision adapts to the illumination provided by the environment so that many blackbody and non-blackbody sources appear white.
- The quality of white light emitted from a blackbody radiator is a function its temperature. This quality is known as color temperature.
- Daylight at midday is often considered the standard value of white light. It produces the same response in the human eye as a blackbody radiator at 6500 K. Visual regions with the same color temperature as midday light often appear neutral white.
- A visual region with a color temperature below 6500 K emits white light that looks reddish in comparison. For cultural reasons light of this color is called warm white even though it is from a "colder" source.
- A visual region with a color temperature above 6500 K emits white light that looks bluish in comparison. For cultural reasons light of this color is called cool white even though it is from a "hotter" source.
- A visual region looks gray if the light from it is similar to white light, but has an intensity somewhat lower than its surroundings.
- Black is the relative absence of visible
light.
- A visual region that emits, reflects, or transmits much less visible light than its surroundings looks black.
- The primary colors of the human visual
system are red, green,
and blue.
- No combination of two primary colors can reproduce a third primary color.
- Combinations of the primary colors will reproduce a wider range of colors than than can be reproduced using any other three colors.
- Combinations of primary colors follow the rules of additive color mixing.
- red + green = yellow
- green + blue = cyan
- blue + red = magenta
- red + green + blue = white
- no light = black
- Systems that work by additive color mixing include …
- photographic and movie film (prints, slides, negatives)
- television and computer displays
- The secondary colors of the human visual system are cyan, magenta, and yellow.
- A complementary color is formed by subtracting a primary color from white light.
- Every secondary color is the complement of a primary color.
- white − red = cyan
- white − green = magenta
- white − blue = yellow
- Every primary color is the complement of a secondary color.
- white − cyan = red
- white − magenta = green
- white − yellow = blue
- Combining complementary colors of light produces light that looks white. As a result, complementary colors are sometimes called opposite colors.
- red + cyan = white
- green + magenta = white
- blue + yellow = white
- Combinations of the secondary colors (pigments) follow the rules of subtractive color mixing.
- cyan + magenta = blue
- magenta + yellow = red
- yellow + cyan = green
- cyan + magenta + yellow = black (though the quality of this black is poor)
- no pigment = white
- Systems that work by subtractive color mixing include …
- three-color printing
- pigment mixing (as in custom paints)
- The "primary colors" of the painter's color wheel are red, yellow, and blue
- When combining paints (or other similar pigment carriers) in equal quantities …
- red + yellow = orange
- yellow + blue = green
- blue + red = purple (which is not the same as violet)
- red + yellow + blue = brown
- no paint = white
- The misidentification of these colors as "primary" is a historical artifact. A greater range of colors can be reproduced using cyan, magenta, and yellow than can be reproduced using red, yellow, and blue.
- Although this is called the painter's color wheel, no serious painter would claim it possible to reproduce every desired color from these three pigments.
- Color Spaces
- All color spaces have at least three dimensions
- RGB (red, green, blue)
- Named for the dominant wavelength of the three light sources used.
- Numbers ranging from zero to some bit number maximum (256, 65536, etc.) are used to describe the relative intensity of each of the three light sources.
- black: none of the light sources are turned on
(R = G = B = 0)
- white: all light sources are turned up as bright as they can
(R = G = B = maximum value)
- CMY (cyan, magenta, yellow)
- Named for the secondary color of the three inks when viewed under white light on a white sheet of paper.
- Numbers ranging from 0% to 100% are used to describe the per cent coverage of a blank sheet of white paper by each of the three inks.
- white: no ink on a white sheet of paper
(C = M = Y = 0%)
- black: paper completely covered with each type of ink
(C = M = Y = 100%)
- HSB (hue, saturation, brightness) a.k.a. HSV (hue, saturation, value)
- The hue angle describes the most visually dominant wavelength
- 000° = red
- 060° = yellow
- 120° = green
- 180° = cyan (similar to, but not the same as the pigment; something like sky blue)
- 240° = blue
- 300° = magenta (similar to, but not the same as the pigment; something like violet or purple)
- 360° = red
- The saturation per cent describes the vibrancy.
- 000% = desaturated (white)
- 100% = saturated (as vibrant as the system will allow)
- The brightness or value per cent describes the relative intensity.
- 000% = dark (black)
- 100% = bright (as bright as the system will allow)
- HSL (hue, saturation, lightness) a.k.a. HSI (hue, saturation, intensity)
- The hue angle describes the most visually dominant wavelength
- Same hue angles as in HSB
- The saturation per cent describes the vibrancy.
- 000% = desaturated (grayscale)
- 100% = saturated (as vibrant as the system will allow)
- The lightness per cent describes the relative intensity.
- 000% = dark (black)
- 050% = medium
- 100% = light (white)
- XYZ (tristimulus)
- Some printing schemes use more than three colors of ink
- CMYK (cyan, magenta, yellow, black)
- CMYK+spot (cyan, magenta, yellow, black, special ink or coating)
- CcMmYK (cyan, light cyan, magenta, light magenta, yellow, black)
- CMYKOG a.k.a. Hexachrome™ (cyan, magenta, yellow, black, orange, green)
- Thin Film Interference
- Resolving Power
- Single, Double, Multiple Source Interference
- Döppler Effect (Light)
- Cerenkov Radiation
- Polarization
- The term polarization refers to the orientation
of the plane of the disturbance in a transverse wave.
- Light and all electromagnetic waves can be polarized.
- Sound and other
longitudinal
waves cannot be polarized.
- A transverse wave is said to have …
- no polarization or to be unpolarized if
its polarization changes quickly and randomly.
- linear polarization if its polarization
does not change; for example …
- vertical
- horizontal
- parallel
- perpendicular
- diagonal
- circular polarization if its polarization rotates.
- elliptical polarization if it has both
linear and circular polarization.
- Methods for tranforming unpolarized light into polarized light include …
- reflection from a dielectric surface
- scattering off the molecules in a fluid (or small particles suspended
in a fluid)
- transmission through a dichroic, crystalline solid
- transmission through a birefringent, crystalline solid
- Geometric Optics
- Reflection
- Optics is the study of the nature and behavior of light.
It can be divided into subdisciplines based on the type of model used to describe light.
- In physical optics, light is assumed to behave like a classical wave.
- In quantum optics, light is assumed to have both wave and particle properties.
- Particles of light are called photons.
- In geometric optics, light is assumed to travel in a definite direction with relatively little diffraction.
- This behavior is known as rectilinear propagation.
- The path of propagation of a light wave is a geometric ray.
- The rays of geometric optics …
- are perpendicular to the wave fronts of physical optics.
- indicate the most probable path of the photons of quantum optics.
- A ray is the path of least action connecting two points in space and is also …
- the path of least time (the quickest path)
- the path of least distance (the shortest path)
- unique and therefore reversible
- The principle of reversibility states that light will follow exactly the same path if its direction of travel is reversed.
- Rays are …
- The eye can see something only if a ray from the object reaches the eye.
- Interface
- An interface is the boundary between …
- two different media.
- two regions of a medium with different characteristics such as …
- density (which is often related to temperature)
- concentration of solute (salinity, for example)
- mechanical stress
- When an incident ray meets an interface it will be partially
- reflected
- Reflected rays obey the law of reflection described in this section of this book.
- transmitted
- Transmitted rays obey Snell's law, which is described in the next section of this book.
- absorbed
- Absorbed rays obey the law of conservation of energy. (The energy of the ray is not destroyed, but changes form.)
- Angles in geometric optics are measured with respect to a line normal to the interface.
- The angle of incidence is the angle between the incident ray and the normal.
- The angle of reflection is the angle between the reflected ray and the normal.
- The angle of refraction is the angle between the transmitted ray and the normal.
- Reflection
- Law of reflection: The angle of incidence equals the angle of reflection.
- The law of reflection can be derived from the principle of least action.
- Refraction
- Refraction is the change in direction
of a wave caused by a change in wave speed.
- An interface is the boundary between
two different media …
- or two regions of a medium with different characteristics such
as …
- density (which is often related to temperature)
- concentration of solute (salinity, for example)
- mechanical stress
- In geometric optics, when an incident ray meets
an interface it will be partially
- reflected
- Reflected rays obey the law of reflection described in
a previous section of this book.
- Materials that reflect a significant portion of incident
light appear shiny or lustrous.
- transmitted
- Transmitted rays obey Snell's law of refraction, which
is described in this section of this book.
- Materials that transmit a significant portion of incident
light appear clear or transparent.
- Materials that do not transmit any incident light are said
to be opaque.
- absorbed
- The energy of absorbed rays is not destroyed, but changes
form.
- Materials that absorb a significant portion of incident
light appear dark.
- Angles are measured with respect to the line normal to the surface.
- The angle of incidence is the
angle between the incident ray and
the normal.
- The angle of reflection is the
angle between the reflected ray and
the normal.
- The angle of refraction is the
angle between the transmitted ray and
the normal.
- Refraction is described mathematically by Snell's law.
where
| |
|
|
| n1 and n2 |
are the indexes of refraction of the two media |
| θ1 and θ2 |
are the angles between the ray and the line
normal to the surface in the two media |
| |
|
|
- Snell's law describes the path of least action between two points
in different media
- The index of refraction …
- is a property of a medium
- is a measure of the "slowness" of a wave
- is defined mathematically by the formula
where
| |
|
|
| n |
is the index of refraction |
| c |
is the speed of light in vacuum |
| v |
is the speed of light in a medium |
| |
|
|
- is always greater than 1 (since the speed of light in a medium
is always slower than the speed of light in a vacuum)
- has no units (since it is the ratio of two speeds)
- generally increases with the density of the medium (and is sometimes
referred to informally as the optical density)
- Total internal reflection occurs when
- Snell's law has no real solution
- light travels from a "slow medium" to a "fast medium" (n1 > n2)
- the incident angle is greater than the critical angle
- The critical angle is the incident angle that corresponds to
a refracted angle of 90° (that is, the transmitted ray travels
parallel to the interface)
- Dispersion
- the speed of light in a medium (and thus the index of refraction)
is a function of frequency and medium
- Birefringence
- Spherical Mirrors
- The magnification formula.
- The spherical mirror formula.
- Spherical Lenses
- A lens is any piece of transparent material
with at least one curved surface.
- There are two types of lenses.
- A converging lens is any piece of
transparent material that will converge parallel rays of light down
to a point.
- Converging lenses are thicker in the middle than at the edges.
- A diverging lens is any piece of transparent
material that will cause parallel rays of light to appear to diverge
from a point.
- Diverging lenses are thicker at the edges than in the middle.
- The focal point or focus is
the point which rays of light initially parallel converge towards after
emerging from a converging lens or diverge from after emerging from a
diverging lens.
- In diagrams, the focus is indicated with the symbol F.
- The magnification formula.
- The thin lens formula (physicist form).
- The thin lens formula (optometrist form).
- Aberration
- For an ideal image-forming optical system …
- there is a one-to-one correspondence between points in the object space
and points in the image space (that is, points map to points not circles,
ellipses or blobs) and
- straight lines in the object space correspond to straight lines in
the image space.
- An aberration is a departure of an image-forming
optical system from ideal behavior.
- Chromatic aberrations are caused by dispersion
(the variation in the index of refraction of a medium with wavelength).
- Longitudinal chromatic aberration (or axial chromatic aberration)
occurs because the image distance varies with wavelength
- Lateral chromatic aberration occurs because the
the image size varies with wavelength.
- Monochromatic aberrations are caused by geometry (the shape of the
lens or mirror).
- Spherical aberration occurs because the focal length of a lens
varies with distance from the principal axis.
- Coma
- Distortion
- Astigmatism
- Field Curvature
prev | up | next
