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Title: Geometrical and Physical Optics. Authors: Longhurst, R S Review_ 2 R S medical-site.info, Articles of Volume 32, kB, Adobe PDF, View/Open. Free download of Geometrical Optics and Physical Optics by Herimanda A. Ramilison. Available in PDF, ePub and Kindle. Read, write reviews and more. Geometrical and Physical Optics by R.S. Longhurst, , available at Book Depository with free delivery worldwide.
The origin is at the vertex. Give also equivalent site written in English Each group will then debrief the tenured teacher or tutor about their results in an attachment sent by e-mail.
This representation will defne a thin lens. This way you will be able to draw the path of a ray of light and predict experimental results. You will need to establish the conjugate relations that will be used to build optical systems later on. Key Concepts Thin Lens: A lens is considered to be thin when its thickness is negligible in relation to the radii of curvature for each face.
Object Focal Length: The distance between the optical center and the object focal point. The point where the rays coming from the object meet up with the optical axis and exit the lens parallel to it. The rays coming from the object traveling parallel to the optical axis exit the lens and meet the axis at the image focal point. Optical Center: It is the point on the axis that for every ray that propagates through the lens and crossing that point there is a parallel emerging ray and an incident ray.
Activity Description This learning activity is made up of problems related to lens characteristics. Chapter 5 Lenses. From these elements we will be able to defne the conditions for using the thin lens representation, and those conditions will amount to all of the characteristics for a thin lens.
Geometrical interpretations have allowed us to draw the path of a ray of light and predict experimental results related to thin lens problems. The results for curved boundaries are quite relevant for the study of lenses since the faces are restricted by curved boundaries. Each face has a radius of curvature equal to 12 cm and their vertices are 1.
The approximation error is: The optical center is: Calculate the image and object focal lengths for a biconvex thin lens. The image focal length is: The object focal length is: They must complete all of the exercises. They will be divided into work groups. Each group must solve the problems and will designate a spo- kesperson. After a certain amount of time 4 hours decided by the tutor, each spokesperson will write up a report, making sure that the frst and last name of each group member is included, and will then send their report by e-mail to the tenured teacher for the course.
Alr car v rlua ur vers ly Z0 Answer Key Question The distances are algebraic values and are defned from left to right. Give the correct algebraic values for the variables used in the equations a.
You have mixed up the variables. Great answer, the distances are indeed algebraic values defned from left to right. Great answer, the distances are indeed algebraic values defned from left to right..
Careful, you have not correctly identifed the variables. Question Careful, this is a diverging lens. You are confused between a diverging and a converging lens. Think it through before answering. Good answer, the lens is divergent and we did not use the thin lens ap- proximation. Question a. Reread the question before answering. Think it through some more. Great answer, the lens is indeed divergent and we consider its thickness to be negligible.
Reread the question before trying again. Alr car v rlua ur vers ly Z1 Question 2. Good answer, You have understood the question. Question 3: O does not depend on the refractive index for the glass that the thick lens is made of. O is positioned between S 1 and S 2 if R 1 and R 2 are of opposite signs. O is always closest to the face with the most curvature. Alr car v rlua ur vers ly Z2 Therefore: Question 4: The optical center O could then merge with the vertices S 1 and S 2 ; the nodal points and principal points will also merge with O.
Good answer Question a. You are mistaken, Try again. Good answer, relation from the course gives the same result. Review your course. You are confused. Good answer d. When those two conditions are satisfed, the optical center O can merge with the vertices S 1 and S 2 ; the nodal points and principal points will also merge with O. Very good, is can see that you understand.
Alr car v rlua ur vers ly Z3 Question 7: A thin lens is a centered system whose principal planes merge with the optical center. Question 8 a. Very good, such a lens is a diverging lens. Think about it some more. Carefully reread the question. Very good, the vergency is negative since the lens is divergent. Question 9: This focal length is related to the through the equation: The expression that relates the vergency optical power V of a thin lens allow us to see that two types of thin lenses can exist: The two focal points are real.
SC will be considered positive if the face is convex and negative if the face si concave. Alr car v rlua ur vers ly Z5 To accomplish this we can use two of the rays that come from B: We are looking for the point when the incident ray intersects the object focal point F s ; the emerging ray will run parallel to F s O. Excellent commentary ; nice precise and clear sketches. Careful, the subject is a biconvex lens b. Careful, we are talking about the image focal length.
Very good, the lens is biconvex and thus the image focal length is nega- tive. Very good, the lens is biconvex and thus the image focal length is posi- tive. Careful, the subject is a biconvex lens d. Conjugate relations for a central origin: Very good, the equations are correct. Careful, the subject is the image plane not the object plane. This is not the right answer. Great answer, you understand. This is not quite the right value.
Be careful, the distance is not negative. Double-check your formula, this is not the right answer. Which fgure s are being referred to in this question? Self Evaluation The learners write down the diffculties they encountered while solving the pro- blems. This will allow them to review the course sections they did not understand as well and to prepare them for the summary evaluation. Required Time 4 hours Specific Objectives To be able to: UCAD of Dakar. Activity Summary For this activity, it is not necessary to complete the sentences.
The goal is for you to read the recommended text so that you may identify the rules for creating certain questions like short simple questions. Alr car v rlua ur vers ly Z9 Formative Evaluation 1.
Recall the rules for creating a simple question. Which of the subsequent sentences follow those rules: Some incident rays meet a boundary with an angle of incidence that is larger than the critical angle and are therefore subject to …………………….
Some ………………… meet a boundary with an angle of incidence that is larger than the critical angle and are therefore subject to total internal refection. Some incident rays meet a boundary with an angle of incidence that is …………………………………….. Learning Activities The learners read the recommended text while focusing on the part that is related to the exercise.
After three hours of reading, they answer the two questions. They then write up an individual report and send it as an attachment in an e-mail to the tutor for the course. Answer Key 1. There are three rules for creating a short question. The space to fll must not be at the beginning of a sentence.
The length of the space must not act as a hint to the answer. The space to fll must always be at the end of a sentence. Only sentence a obeys the rules. Self Evaluation This evaluation allows the learners to realize the diffculty in creating an evaluation and to be aware of the way they should choose evaluation tools later on.
Teacher Guidelines This evaluation is optional and is therefore not mandatory. Only those who wish will complete it. The tutor will correct their work and can ask them to make up some simple questions.
The grade for this activity will not be considered in the fnal evaluation. Alr car v rlua ur vers ly 80 XV. Module synthesis This training module is made up of a prerequisite exam and four Leaning Units, among other things. The prerequisite test is mandatory for all of the learners; the terms of success are included.
Each Learning Unit includes specifc objectives and a learning activity. In each learning activity there is recommended reading, useful links, resources and corrected problems. Each exercise assesses the specifc learning objectives for one of the units. All of the steps that the students must take to solve the problems are included. The useful links presented in the module as well as in the learning activities are websites that are related to the content that the students must master. It is strongly recommended that the students consult them.
Just like the readings and resources, they are helpful with the learning process. In the module you will also fnd an optional educational formative evaluation. Just like its name indicates, it is not mandatory but is nonetheless necessary for future teachers. A summary evaluation will close the different evaluations in this module. Alr car v rlua ur vers ly 81 XVI.
A medium that has the same properties in every point is said to be ……………. A medium whose properties observed at a certain point do not depend on the direction of observation is said to be …………… c. In a homogenous and i sot ropi c medi um, l i ght propagat es ………………. By defnition: Choose the right answer among the following equations: Complete the following sentence: The surface of two homogenous isotropic media constitutes ………………… 5.
A teacher presents the following rules to his students as the Snell-Des- cartes Laws.
Which of the following propositions is correct? The angles of incidence and of refection are equal and in opposite directions to one another. For every monochromatic light, the sines of the angle of incidence and of refraction are joined by the relation: The refected ray and the refracted ray are on the plane of inci- dence.
A medium 1 with refractive index n 1 is less refractive than a medium 2 with refractive index n 2 means that: A plane boundary is only rigorously stigmatic in two situations: When the point source is at infnity; its pinpoint image is itself at inf- nity. False Circle the right answer. A system is stigmatic if for a certain object point there is a unique image point.
False A plane mirror is a fat surface capable of: Choose the right answer. The ratio between the focal lengths for a curved boundary is equal to: The center of a converging boundary is always found: Alr car v rlua ur vers ly 81 The center of a diverging boundary is always found: Vergency optical power is expressed: Lenses that are thin at the periphery are: Lenses that are thick at the periphery are: For a converging lens: Which of these assertions are false?
An oscillator is a system whose state is described by a variable x. Alr car v rlua ur vers ly 85 The pseudo-period describes the period of an anharmonic oscillator.
The initial conditions are optional when studying an oscillator. The oscillations are said to be damped when the external force applied to the system to set it in motion is very brief, and disappears as soon as the system starts to oscillate. Free oscillation is characterized by: For a damped harmonic oscillator, it is possible for there to be no more oscillation. A damped harmonic oscillator is characterized by: Match up the following two columns by associating each number with the right letter.
The logarithmic decrement measures the decrease in: The quality factor Q and the viscous damping coeffcient l are: Alr car v rlua ur vers ly 8Z The quality factor Q and the natural frequency w 0 characterize the dam- ped harmonic oscillator. Answer Key 1a. Good answer, 1b. Very good 2. Try again, n is defnitely not the product of two speeds.
Think about it before answering. The period is not the reciprocal of the wavelength.
The period is not equal to the product of the wavelength and the fre- quency. Good answer, the period is indeed the reciprocal of the frequency. The period is not equal to the reciprocal of the pulse.
Very good, you understand well the defnition of a boundary. A teacher presents the following rules to his students as the Snell-Descartes Laws.
This is not the frst law. This is not the second law. This is not the third law. Too bad, this is not the right order. This is not the right order.
This is not the right order, give it some thought. This is not the right order, try again. Alr car v rlua ur vers ly 88 7. There is no reason for one of the indexes to be double the other. Why half? Good job, these assertions are correct.
Carefully reread the two situations to identify the right one. Good answer, this is the defnition of astigmatism. You do not seem to know the defnition of astigmatism. Good job, mirrors do indeed refect light. A plane mirror never refracts light. A plane mirror never diffracts light. A plane mirror never absorbs light. The focal lengths are not proportional to the refractive indexes.
Good answer, the ratio between the focal lengths is indeed equal to the product of the refractive indexes. Reread the question. As a matter of fact, if it were otherwise, the rays could not converge. If the rays converge, the boundary cannot be convergent. Good answer, for a converging boundary, the center must be in the most refractive medium. Careful, the boundary is converging not diverging.
Alr car v rlua ur vers ly 89 Excellent answer, it must always be in a less refractive medium. Very good, the diopter is indeed the unit used for vergency optical power. Vergency optical power is not an abstract number. Diverging is indeed the expression associated with diverging lenses. Very good, a thin periphery is used in converging lenses. Good answer, thick periphery goes with diverging. Converging is used for thin lenses. Careful, this assertion is actually correct, it is not at all false.
Good answer, this ray is indeed neither deviated or refected. Careful, this assertion is not false, it is true. Good answer, the oscillator does indeed oscillate for a certain value. Careful, reread the question. Good answer, you have understood the concept. A harmonic oscillator is a system whose time describing variable is a sinusoidal function of time.
Good answer, you have good knowledge of oscillators. You have not made the right choice. Alr car v rlua ur vers ly 90 You were defnitely in a rush to answer.
Good answer, the pseudo-period is not used for an anharmonic oscillator. Review the equation for the period: Good answer, on the contrary! You have defnitely misunderstood the question.
Good job, it is rather valid for free oscillations. Good answer, all oscillations have amplitude. Careful, an oscillation is not characterized by a wavelength. Very good, every oscillation has its own natural frequency. Good job, every oscillation has phase j. Good answer, this is the case for strong damping. Good answer, the viscous damping coeffcient l does characterize the harmonic oscillator.
Careful, the period T 0 does not characterize the harmonic oscillator. Good job, the natural frequency w 0 does characterize the harmonic os- cillator. Phase j does not characterize the harmonic oscillator. Alr car v rlua ur vers ly 91 Good answer, you understand the meaning of aperiodical. Good job, you understand the meaning of critical speed.
For each of these answers: Review your course, you have misunderstood the terms. Good answer, the logarithmic decrement does indeed measure the decrease in amplitude. Try again, the viscous damping coeffcient l is not related to the logarithmic decrement. Careful, try to remember the quality factor formula. Very good, the quality factor is indeed inversely proportional to the viscous damping coeffcient.
Careful, they cannot be independent from one another, they are linked. Good answer, the quality factor does indeed depend on these two factors. Take your time answering. Alr car v rlua ur vers ly 92 XVII. Mada- gascar. UCAD de Dakar. Evaluation continue et examens. Editions Labor. Education Editions Nouvelles. Corriger des copies. Evaluer pour former. Reflection is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated.
Common examples include the reflection of light, sound and water waves. The law of reflection says that for specular reflection the angle at which the wave is incident on the surface equals the angle at which it is reflected. Mirrors exhibit specular reflection. In acoustics, reflection causes echoes and is used in sonar. In geology, it is important in the study of seismic waves. Reflection is observed with surface waves in bodies of water.
Reflection is observed with many types of electromagnetic wave, besides visible light. Reflection of VHF and higher frequencies is important for radio transmission and for radar. Even hard X-rays and gamma rays can be reflected at shallow angles with special "grazing" mirrors.
The sun is reflected in the water, which is reflected in the paddle. Furthermore, if the interface is between a dielectric and a conductor, the phase of the reflected wave is retained, otherwise if the interface is between two dielectrics, the phase may be retained or inverted, depending on the indices of refraction. Reflection is enhanced in metals by suppression of wave propagation beyond their skin depths.
Reflection also occurs at the surface of transparent media, such as water or glass. Diagram of specular reflection In the diagram at left, a light ray PO strikes a vertical mirror at point O, and the reflected ray is OQ. An Indian triggerfish reflecting in the water surface through total internal reflection. In fact, reflection of light may occur whenever light travels from a medium of a given refractive index into a medium with a different refractive index.
In the most general case, a certain fraction of the light is reflected from the interface, and the remainder is refracted. Solving Maxwell's equations for a light ray striking a boundary allows the derivation of the Fresnel equations, which 6 can be used to predict how much of the light reflected, and how much is refracted in a given situation. Total internal reflection of light from a denser medium occurs if the angle of incidence is above the critical angle.
Total internal reflection is used as a means of focussing waves that cannot effectively be reflected by common means. X-ray telescopes are constructed by creating a converging "tunnel" for the waves. As the waves interact at low angle with the surface of this tunnel they are reflected toward the focus point or toward another interaction with the tunnel surface, eventually being directed to the detector at the focus. A conventional reflector would be useless as the X-rays would simply pass through the intended reflector.
When light reflects off a material denser with higher refractive index than the external medium, it undergoes a polarity inversion. In contrast, a less dense, lower refractive index material will reflect light in phase. This is an important principle in the field of thin-film optics.
Specular reflection forms images. Reflection from a flat surface forms a mirror image, which appears to be reversed from left to right because we compare the image we see to what we would see if we were rotated into the position of the image. Specular reflection at a curved surface forms an image which may be magnified or demagnified; curved mirrors have optical power.
Such mirrors may have surfaces that are spherical or parabolic. Specular reflection If the reflecting surface is very smooth, the reflection of light that occurs is called specular or regular reflection. The laws of reflection are as follows: The incident ray, the reflected ray and the normal to the reflection surface at the point of the incidence lie in the same plane. The angle which the incident ray makes with the normal is equal to the angle which the reflected ray makes to the same normal.
Light paths are reversible. Diffuse reflection When light strikes a rough or granular surface, it bounces off in all directions due to the microscopic irregularities of the interface. Thus, an 'image' is not formed. This is called diffuse reflection. The exact form of the reflection depends on the structure of the surface. One common model for diffuse reflection is Lambertian reflectance, in which the light is reflected with equal luminance in photometry or radiance in radiometry in all directions, as defined by Lambert's cosine law.
Retroreflector Some surfaces exhibit retroreflection. The structure of these surfaces is such that light is returned in the direction from which it came. When flying over clouds illuminated by sunlight the region seen around the aircraft's shadow will appear brighter, and a similar effect may be seen from dew on grass. This partial retro- 8 reflection is created by the refractive properties of the curved droplet's surface and reflective properties at the backside of the droplet. Some animal's retinas act as retroreflectors, as this effectively improves the animal's night vision.
Since the lenses of their eyes modify reciprocally the paths of the incoming and outgoing light the effect is that the eyes act as a strong retroreflector, sometimes seen at night when walking in wildlands with a flashlight. A simple retroreflector can be made by placing three ordinary mirrors mutually perpendicular to one another a corner reflector. The image produced is the inverse of one produced by a single mirror.
A surface can be made partially retroreflective by depositing a layer of tiny refractive spheres on it or by creating small pyramid like structures. In both cases internal reflection causes the light to be reflected back to where it originated.
This is used to make traffic signs and automobile license plates reflect light mostly back in the direction from which it came. In this application perfect retroreflection is not desired, since the light would then be directed back into the headlights of an oncoming car rather than to the driver's eyes. In this type of reflection, not only the direction of the light is reversed, but the actual wavefronts are reversed as well.
A conjugate reflector can be used to remove aberrations from a beam by reflecting it and then passing the reflection through the aberrating optics a second time. In the physical and biological sciences, the reflection of neutrons off atoms within a material is commonly used to determine its internal structures. Note that audible sound has a very wide frequency range from 20 to about Hz , 9 and thus a very wide range of wavelengths from about 20 mm to 17 m.
As a result, the overall nature of the reflection varies according to the texture and structure of the surface. For example, porous materials will absorb some energy, and rough materials where rough is relative to the wavelength tend to reflect in many directions—to scatter the energy, rather than to reflect it coherently.
This leads into the field of architectural acoustics, because the nature of these reflections is critical to the auditory feel of a space. In the theory of exterior noise mitigation, reflective surface size mildly detracts from the concept of a noise barrier by reflecting some of the sound into the opposite direction. Study of the deep reflections of waves generated by earthquakes has allowed seismologists to determine the layered structure of the Earth.
Shallower reflections are used in reflection seismology to study the Earth's crust generally, and in particular to prospect for petroleum and natural gas deposits.
All of these waves add up to specular reflection following Hero's equi-angular reflection law and refraction. Light—matter interaction in terms of photons is a topic of quantum electrodynamics, and is described in detail by Richard Feynman in his popular book QED: The Strange Theory of Light and Matter.
Refraction An image of the Golden Gate Bridge is refracted and bent by many differing three dimensional pools of water 10 Refraction in a Perspex acrylic block. Refraction is the change in direction of a wave due to a change in its speed.
This is most commonly observed when a wave passes from one medium to another at an angle.
Refraction of light is the most commonly observed phenomenon, but any type of wave can refract when it interacts with a medium, for example when sound waves pass from one medium into another or when water waves move into water of a different depth. In general, the incident wave is partially refracted and partially reflected; the details of this behavior are described by the Fresnel equations. The dark rectangle represents the actual position of a pencil sitting in a bowl of water.
The light rectangle represents the apparent position of the pencil. Notice that the end X looks like it is at Y , a position that is considerably shallower than X. Photograph of refraction of waves in a ripple tank Diagram of refraction of water waves 12 In optics, refraction occurs when light waves travel from a medium with a given refractive index to a medium with another at an angle. At the boundary between the media, the wave's phase velocity is altered, usually causing a change in direction.
Its wavelength increases or decreases but its frequency remains constant. For example, a light ray will refract as it enters and leaves glass, assuming there is a change in refractive index. A ray traveling along the normal perpendicular to the boundary will change speed, but not direction. Refraction still occurs in this case. Understanding of this concept led to the invention of lenses and the refracting telescope. Refraction can be seen when looking into a bowl of water.
Air has a refractive index of about 1. If a person looks at a straight object, such as a pencil or straw, which is placed at a slant, partially in the water, the object appears to bend at the water's surface. This is due to the bending of light rays as they move from the water to the air. Once the rays reach the eye, the eye traces them back as straight lines lines of sight.
The lines of sight shown as dashed lines intersect at a higher position than where the actual rays originated. This causes the pencil to appear higher and the water to appear shallower than it really is. The depth that the water appears to be when viewed from above is known as the apparent depth. This is an important consideration for spearfishing from the surface because it will make the target fish appear to be in a different place, and the fisher must aim lower to catch the fish.
The diagram on the right shows an example of refraction in water waves. Ripples travel from the left and pass over a shallower region inclined at an angle to the wavefront. The waves travel more slowly in the shallower water, so the wavelength decreases and the wave bends at the boundary. The dotted line represents the normal to the boundary. The dashed line represents the original direction of the waves. This phenomenon explains why waves on a shoreline tend to strike the shore close to a perpendicular angle.
As the waves travel from deep water into shallower water near the shore, they are refracted from their original direction of travel to an angle more normal to the shoreline. Glass has a higher refractive index than air. When a beam of white light passes from air into a material having an index of refraction that varies with frequency, a phenomenon known as dispersion occurs, in which different coloured components of the white light are refracted at different angles, i.
The different colors correspond to different frequencies. While refraction allows for beautiful phenomena such as rainbows, it may also produce peculiar optical phenomena, such as mirages and Fata Morgana.
These are caused by the change of the refractive index of air with temperature. Recently some metamaterials have been created which have a negative refractive index. With metamaterials, we can also obtain total refraction phenomena when the wave impedances of the two media are matched.
There is then no reflected wave. First, as light is entering a drop of water, it slows down. If the water's surface is not flat, then the light will be bent into a new path. This round shape will bend the light outwards and as it spreads out, the image you see gets larger.
The marchers on the side that runs into the mud first will slow down first. This causes the whole band to pivot slightly toward the normal make a smaller angle from the normal.
A series of test lenses in graded optical powers or focal lengths are presented to determine which provide the sharpest, clearest vision. The amount of ray bending is dependent upon the amount of difference between sound speeds, that is, the variation in temperature, salinity, and pressure of the water. The phenomenon of refraction of sound in the atmosphere has been known for centuries;  however, beginning in the early s, widespread analysis of this effect came into vogue through the designing of urban highways and noise barriers to address the meteorological effects of bending of sound rays in the lower atmosphere.
The law says that the ratio of the sines of the angles of incidence and of refraction is a constant that depends on the media. In optics, the law is used in ray tracing to compute the angles of incidence or refraction, and in experimental optics and gemology to find the refractive index of a material.
Snell's law is also satisfied in the metamaterials which allow light to be bent "backward" at a negative index, with a negative angle of refraction. Named after Dutch mathematician Willebrord Snellius, one of its discoverers, Snell's law states that the ratio of the sines of the angles of incidence and refraction is equivalent to the ratio of velocities in the two media, or equivalent to the opposite ratio of the indices of refraction: Ptolemy, of the Thebaid, had found a relationship regarding refraction angles, but which was inaccurate for angles that were not small.
Ptolemy was confident he had found an accurate empirical law, partially as a result of fudging his data to fit theory see: It was rediscovered by Thomas Harriot in ,  who however did not publish his results although he had corresponded with Kepler on this very subject.
In , Willebrord Snellius Snel derived a mathematically equivalent form, that remained unpublished during his lifetime. Rejecting Descartes' solution, Pierre de Fermat arrived at the same solution based solely on his principle of least time.
According to Dijksterhuis  , "In De natura lucis et proprietate Isaac Vossius said that Descartes had seen Snell's paper and concocted his own proof. We now know this charge to be undeserved but it has been adopted many times since. Although he spelled his name "Snel", as noted above, it has conventionally been spelled "Snell" in English, apparently by misinterpreting the Latin form of his name, "Snellius".
The indices of refraction of the media, labeled n 1 ,n 2 and so on, are used to represent the factor by which a light ray's speed decreases when traveling through a refractive medium, such as glass or water, as opposed  to its velocity in a vacuum.
These angles are measured with respect to the normal line, represented perpendicular to the boundary. In the case of light traveling from air into water, light would be refracted towards the normal line, because the light is slowed down in water; light traveling from water to air would refract away from the normal line. Refraction between two surfaces is also referred to as reversible because if all conditions were identical, the angles would be the same for light propagating in the opposite direction.
Snell's law is generally true only for isotropic or specular media such as glass. In anisotropic media such as some crystals, birefringence may split the refracted ray into two rays, the ordinary or o-ray which follows Snell's law, and the other extraordinary or e-ray which may not be co- planar with the incident ray.
Main article: Total internal reflection When light travels from a medium with a higher refractive index to one with a lower refractive index, Snell's law seems to require in some cases whenever the angle of incidence is large enough that the sine of the angle of refraction be greater than one. This of course is impossible, and the light in such cases is completely reflected by the boundary, a phenomenon known as total internal reflection.
The largest possible angle of incidence which still results in a refracted ray is called the critical angle; in this case the refracted ray travels along the boundary between the two media. The refractive indices of water and air are approximately 1.
The region below the grey line has a higher index of refraction, and proportionally lower speed of light, than the region above it. Snell's law may be derived from Fermat's principle, which states that the light travels the path which takes the least time. By taking the derivative of the optical path length, the stationary point is found giving the path taken by the light though it should be noted that the result does not show light taking the least time path, but rather one that is stationary with respect to small variations as there are cases where light actually takes the greatest time path, as in a spherical mirror.
In a classic analogy, the area of lower refractive index is replaced by a beach, the area of higher refractive index by the sea, and the fastest way for a rescuer on the beach to get to a drowning person in the sea is to run along a path that follows Snell's law.
Alternatively, Snell's law can be derived using interference of all possible paths of light wave from source to observer—it results in destructive interference everywhere except extrema of phase where interference is constructive —which become actual paths. A homogeneous surface perpendicular to say the z direction can not change the transverse momentum. Since the propagation vector is proportional to the photon's momentum, the transverse propagation direction k x ,k y ,0 must remain the same in both regions.
Using the well known dependence of the wave number on the refractive index of the medium, we derive Snell's law immediately. Note that no surface is truly homogeneous, in the least at the atomic scale. Yet full translational symmetry is an excellent approximation whenever the region is homogeneous on the scale of the light wavelength.
Otherwise, use Example: In this case, an evanescent wave is produced, which rapidly decays from the surface into the second medium. Conservation of energy is maintained by the circulation of energy across the boundary, averaging to zero net energy transmission.
Dispersion optics In many wave-propagation media, wave velocity changes with frequency or wavelength of the waves; this is true of light propagation in most transparent substances other than a vacuum. These media are called dispersive. The result is that the angles determined by Snell's law also depend on frequency or wavelength, so that a ray of mixed wavelengths, such as white light, will spread or disperse. Such dispersion of light in glass or water underlies the origin of rainbows, in which different wavelengths appear as different colors.
In optical instruments, dispersion leads to chromatic aberration; a color-dependent blurring that sometimes is the resolution-limiting effect.
This was especially true in refracting telescopes, before the invention of achromatic objective lenses. Curved mirror From Wikipedia, the free encyclopedia Jump to: The photographer is seen reflected at top right A curved mirror is a mirror with a curved reflective surface, which may be either convex bulging outward or concave bulging inward.
Most curved mirrors have surfaces that are shaped like part of a sphere,  but other shapes are sometimes used in optical devices. The most common non-spherical type are parabolic reflectors, found in optical devices such as reflecting telescopes that need to image distant objects, since spherical mirror systems suffer from spherical aberration. Convex mirrors reflect light outwards, therefore they are not used to focus light.
Such mirrors always form a virtual image, since the focus F and 22 the centre of curvature 2F are both imaginary points "inside" the mirror, which cannot be reached. Therefore images formed by these mirrors cannot be taken on screen.
As they are inside the mirror A collimated parallel beam of light diverges spreads out after reflection from a convex mirror, since the normal to the surface differs with each spot on the mirror. These features make convex mirrors very useful: The passenger-side mirror on a car is typically a convex mirror.
In some countries, these are labelled with the safety warning "Objects in mirror are closer than they appear", to warn the driver of the convex mirror's distorting effects on distance perception. Convex mirrors are used in some automated teller machines as a simple and handy security feature, allowing the users to see what is happening behind them.
Similar devices are sold to be attached to ordinary computer monitors. A concave mirror, or converging mirror, has a reflecting surface that bulges inward away from the incident light.
Concave mirrors reflect light inward to one focal point, therefore they are used to focus light. Unlike convex mirrors, concave mirrors show different image types depending on the distance between the object and the mirror. These mirrors are called "converging" because they tend to collect light that falls on them, refocusing parallel incoming rays toward a focus.
This is because the light is reflected at different angles, since the normal to the surface differs with each spot on the mirror.
Note that the reflected light rays are parallel and do not meet the others. In this way, no image is formed or more properly the image is formed at infinity.
These are the simplest to make, and it is the best shape for general-purpose use. Spherical mirrors, however, suffer from spherical aberration. Parallel rays reflected from such mirrors do not focus to a single point. For parallel rays, such as those coming from a very distant object, a parabolic reflector can do a better job. Such a mirror can focus incoming parallel rays to a much smaller spot than a spherical mirror can. See also: Toroidal reflector  Analysis  Mirror equation and magnification The Gaussian mirror equation relates the object distance d o and image distances d i to the focal length f: The negative sign in this equation is used as a convention.
By convention, if the magnification is positive, the image is upright. If the magnification is negative, the image is inverted upside down. Ray tracing physics The image location and size can also be found by graphical ray tracing, as illustrated in the figures above. A ray drawn from the top of the object to the surface vertex where the optical axis meets the mirror will form an angle with that axis. The reflected ray has the same angle to the axis, but is below it See Specular reflection.
A second ray can be drawn from the top of the object passing through the focal point and reflecting off the mirror at a point somewhere below the optical axis. Such a ray will be reflected from the mirror as a ray parallel to the optical axis. The point at which the two rays described above meet is the image point corresponding to the top of the object.
Its distance from the axis defines the height of the image, and its location along the axis is the image location. The mirror equation and magnification equation can be derived geometrically by considering these two rays. Ray transfer matrix analysis The mathematical treatment is done under the paraxial approximation, meaning that under the first approximation a spherical mirror is a parabolic reflector.
The ray matrix of a spherical mirror is shown here for the concave reflecting surface of a spherical mirror. The C element of the matrix is , where f is the focal point of the optical device. Box 2 shows the Maclaurin series of up to order 1. The derivations of the ray matrices of a convex spherical mirror and a thin lens are very similar.
Brewster's angle From Wikipedia, the free encyclopedia Jump to: Brewster's angle also known as the polarization angle is an angle of incidence at which light with a particular polarization is perfectly transmitted through a surface, with no reflection.
The angle at which this occurs is named after the Scottish physicist, Sir David Brewster — At one particular angle of incidence, however, light with one particular polarization cannot be reflected. The polarization that cannot be reflected at this angle is the polarization for which the electric field of the light waves lies in the same plane as the incident ray and the surface normal i. Light with this polarization is said to be p-polarized, because it is parallel to the plane.
Light with the perpendicular polarization is said to be s-polarized, from the German senkrecht—perpendicular. When unpolarized light strikes a surface at Brewster's angle, the reflected light is always s-polarized.
Although 's' and 'p' polarization states were not named for this convention, it may be convenient to remember that 's' polarized light will "skip" off a Brewster boundary and 'p' polarized light will "plunge" through. The physical mechanism for this can be qualitatively understood from the manner in which electric dipoles in the media respond to p-polarized light. One can imagine that light incident on the surface is absorbed, and then reradiated by oscillating electric dipoles at the interface between the two media.
The polarization of freely propagating light is always perpendicular to the direction in which the light is travelling. The dipoles that produce the transmitted refracted light oscillate in the polarization direction of that light. These same oscillating dipoles also generate the reflected light. However, dipoles do not radiate any energy in the direction along which they oscillate.
Consequently, if the direction of the refracted light is perpendicular to the direction in which the light is predicted to be specularly reflected, the dipoles will not create any reflected light. Since, by definition, the s-polarization is parallel to the interface, the corresponding oscillating dipoles will always be able to radiate in the specular-reflection direction.
This is why there is no Brewster's angle for s-polarized light. Rearranging, we get: This equation is known as Brewster's law. Note that, since all p-polarized light is refracted i. A glass plate or a stack of plates placed at Brewster's angle in a light beam can thus be used as a polarizer.
Since the refractive index for a given medium changes depending on the wavelength of light, Brewster's angle will also vary with wavelength.
The phenomenon of light being polarized by reflection from a surface at a particular angle was first observed by Etienne-Louis Malus in He attempted to relate the polarizing angle to the refractive index of the material, but was frustrated by the inconsistent quality of glasses available at that time. In , Brewster experimented with higher-quality materials and showed that this angle was a function of the refractive index, defining Brewster's law.
Although Brewster's angle is generally presented as a zero-reflection angle in textbooks from the late s onwards, it truly is a polarizing angle. The concept of a polarizing angle can be extended to the concept of a Brewster wavenumber to cover planar interfaces between two linear bianisotropic materials. In a large range of angles around Brewster's angle the reflection of p-polarized light is lower than s-polarized light.
Thus, if the sun is low in the sky reflected light is mostly s-polarized. Polarizing sunglasses use a polarizing material such as polaroid film to block horizontally-polarized light, preferentially blocking reflections from horizontal surfaces.
The effect is strongest with smooth surfaces such as water, but reflections from road and the ground are also reduced. Photographers use the same principle to remove reflections from water so that they can photograph objects beneath the surface.
In this case, the polarizing filter camera attachment can be rotated to be at the correct angle see figure. Photographs taken of mudflats with a camera polarizer filter rotated to two different angles. Since the window reflects some s-polarized light but no p-polarized light, the gain for the s 31 polarization is reduced but that for the p polarization is not affected.
This causes the laser's output to be p polarized, and allows lasing with no loss due to the window. It is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings. Similar effects are observed when light waves travel through a medium with a varying refractive index or a sound wave through one with varying acoustic impedance.
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