English 102 Reflection

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English 102 Reflection



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English 102 Final Reflection

There are two sets of Fresnel coefficients for two different linear polarization components of the incident wave. Since any polarization state can be resolved into a combination of two orthogonal linear polarizations, this is sufficient for any problem. Likewise, unpolarized or "randomly polarized" light has an equal amount of power in each of two linear polarizations. The s polarization refers to polarization of a wave's electric field normal to the plane of incidence the z direction in the derivation below ; then the magnetic field is in the plane of incidence. The p polarization refers to polarization of the electric field in the plane of incidence the xy plane in the derivation below ; then the magnetic field is normal to the plane of incidence.

In the diagram on the right, an incident plane wave in the direction of the ray IO strikes the interface between two media of refractive indices n 1 and n 2 at point O. Part of the wave is reflected in the direction OR , and part refracted in the direction OT. The relationship between these angles is given by the law of reflection :. The behavior of light striking the interface is solved by considering the electric and magnetic fields that constitute an electromagnetic wave , and the laws of electromagnetism , as shown below. The ratio of waves' electric field or magnetic field amplitudes are obtained, but in practice one is more often interested in formulae which determine power coefficients, since power or irradiance is what can be directly measured at optical frequencies.

The power of a wave is generally proportional to the square of the electric or magnetic field amplitude. We call the fraction of the incident power that is reflected from the interface the reflectance or "reflectivity", or "power reflection coefficient" R , and the fraction that is refracted into the second medium is called the transmittance or "transmissivity", or "power transmission coefficient" T. Note that these are what would be measured right at each side of an interface and do not account for attenuation of a wave in an absorbing medium following transmission or reflection. The reflectance for s-polarized light is. We assume that the media are non-magnetic i. Making this substitution, we obtain equations using the refractive indices:.

Note that all such intensities are measured in terms of a wave's irradiance in the direction normal to the interface; this is also what is measured in typical experiments. Although these relationships describe the basic physics, in many practical applications one is concerned with "natural light" that can be described as unpolarized. That means that there is an equal amount of power in the s and p polarizations, so that the effective reflectivity of the material is just the average of the two reflectivities:. For low-precision applications involving unpolarized light, such as computer graphics , rather than rigorously computing the effective reflection coefficient for each angle, Schlick's approximation is often used.

Thus, the reflectance simplifies to. At a dielectric interface from n 1 to n 2 , there is a particular angle of incidence at which R p goes to zero and a p-polarised incident wave is purely refracted, thus all reflected light is s-polarised. When light travelling in a denser medium strikes the surface of a less dense medium i. The above equations relating powers which could be measured with a photometer for instance are derived from the Fresnel equations which solve the physical problem in terms of electromagnetic field complex amplitudes , i.

Those underlying equations supply generally complex-valued ratios of those EM fields and may take several different forms, depending on formalisms used. The complex amplitude coefficients are usually represented by lower case r and t whereas the power coefficients are capitalized. In the following, the reflection coefficient r is the ratio of the reflected wave's electric field complex amplitude to that of the incident wave. The transmission coefficient t is the ratio of the transmitted wave's electric field complex amplitude to that of the incident wave.

We require separate formulae for the s and p polarizations. Note that in the cases of an interface into an absorbing material where n is complex or total internal reflection, the angle of transmission might not evaluate to a real number. We consider the sign of a wave's electric field in relation to a wave's direction. Consequently, for p polarization at normal incidence, the positive direction of electric field for an incident wave to the left is opposite that of a reflected wave also to its left ; for s polarization both are the same upward. Using these conventions, [5] [6]. One can write similar equations applying to the ratio of magnetic fields of the waves, but these are usually not required. On the other hand, calculation of the power transmission coefficient T is less straightforward, since the light travels in different directions in the two media.

What's more, the wave impedances in the two media differ; power is only proportional to the square of the amplitude when the media's impedances are the same as they are for the reflected wave. This results in: [9]. In the case of total internal reflection where the power transmission T is zero, t nevertheless describes the electric field including its phase just beyond the interface. The phase shift of the reflected wave on total internal reflection can similarly be obtained from the phase angles of r p and r s whose magnitudes are unity. These phase shifts are different for s and p waves, which is the well-known principle by which total internal reflection is used to effect polarization transformations.

When light makes multiple reflections between two or more parallel surfaces, the multiple beams of light generally interfere with one another, resulting in net transmission and reflection amplitudes that depend on the light's wavelength. The interference, however, is seen only when the surfaces are at distances comparable to or smaller than the light's coherence length , which for ordinary white light is few micrometers; it can be much larger for light from a laser. An example of interference between reflections is the iridescent colours seen in a soap bubble or in thin oil films on water. A quantitative analysis of these effects is based on the Fresnel equations, but with additional calculations to account for interference.

In , the dependence of the polarizing angle on the refractive index was determined experimentally by David Brewster. In , however, Augustin-Jean Fresnel derived results equivalent to his sine and tangent laws above , by modeling light waves as transverse elastic waves with vibrations perpendicular to what had previously been called the plane of polarization. Details of Fresnel's derivation, including the modern forms of the sine law and tangent law, were given later, in a memoir read to the French Academy of Sciences in January In the same memoir of January , [24] Fresnel found that for angles of incidence greater than the critical angle, his formulas for the reflection coefficients r s and r p gave complex values with unit magnitudes.

Noting that the magnitude, as usual, represented the ratio of peak amplitudes, he guessed that the argument represented the phase shift, and verified the hypothesis experimentally. The success of the complex reflection coefficient inspired James MacCullagh and Augustin-Louis Cauchy , beginning in , to analyze reflection from metals by using the Fresnel equations with a complex refractive index. Thus Fresnel's interpretation of the complex values of his reflection coefficients marked the confluence of several streams of his research and, arguably, the essential completion of his reconstruction of physical optics on the transverse-wave hypothesis see Augustin-Jean Fresnel. In order to compute meaningful Fresnel coefficients, we must assume that the medium is approximately linear and homogeneous.

In optics, one usually knows the refractive index n of the medium, which is the ratio of the speed of light in a vacuum c to the speed of light in the medium. In the analysis of partial reflection and transmission, one is also interested in the electromagnetic wave impedance Z , which is the ratio of the amplitude of E to the amplitude of H. The last-mentioned relation, however, will make it convenient to derive the reflection coefficients in terms of the wave admittance Y , which is the reciprocal of the wave impedance Z.

In the case of uniform plane sinusoidal waves, the wave impedance or admittance is known as the intrinsic impedance or admittance of the medium. This case is the one for which the Fresnel coefficients are to be derived. In a uniform plane sinusoidal electromagnetic wave , the electric field E has the form. So a phase advance is equivalent to multiplication by a complex constant with a negative argument. That factor also implies that differentiation w. Solving for k gives. The electric field for a uniform plane sine wave will then be represented by the location-dependent phasor. Taking the magnitude equations and substituting from 2 , we obtain. Multiplying the last two equations gives.

Dividing the second result by the first gives. Then the xz plane is the interface, and the y axis is normal to the interface see diagram. Let i and j in bold roman type be the unit vectors in the x and y directions, respectively. Hence, by 2 , the magnitude of the wave vector is proportional to the refractive index. From the magnitudes and the geometry, we find that the wave vectors are. The corresponding dot products in the phasor form 3 are. For the s polarization, the E field is parallel to the z axis and may therefore be described by its component in the z direction. Then, if the incident E field is taken to have unit amplitude, the phasor form 3 of its z component is.

Under the sign convention used in this article, a positive reflection or transmission coefficient is one that preserves the direction of the transverse field, meaning in this context the field normal to the plane of incidence. The paper is about making the choice to wear a motorcycle helmet. This paper falls into the category of ethical and responsible behavior because, in Indiana, riders have a choice to wear their helmet. Choosing to wear a helmet would be responsible behavior and choosing to wear a novelty helmet that is not DOT approved in a state that requires certified helmets would be unethical. Another unethical example would be for.

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Even before the semester started the feeling of anxiety had taken over, as writing has never been my strongest skill, but stayed positive and confident I would be successful. Much to my surprise, my instructor, Michael Critchfield quickly addressed that this class would not be easy and amazingly changed my views of writing throughout the entire semester. During this semester, the realization of my past was that I have often skimmed through parts of the writing process to get the job done. Not being an avid reader often hindered my writing abilities.

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