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Reflection & Refraction
Optics 430/530, week III Reflection & Refraction This class notes freely use material from P. Piot, PHYS , NIU FA2018
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Refraction at an interface
Assume an incident plane wave propagating in medium with index of refraction 𝑛 𝑖 and ente- ring a medium with index 𝑛 𝑡 Two waves results at from the interface i/t: Transmitted wave which propagates in medium 𝑛 𝑡 Reflected wave which reflect from the interface and propagates in 𝑛 𝑡 We consider the media to be isotropic so wave vectors associated with the three waves are all in the same plane (known as plane of incidence) P. Piot, PHYS , NIU FA2018
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Perpendicular to plane of incidence
s and p waves P- 𝑥𝑥̂ It is convenient to decompose the electric field vector into two orthogonal component Consider the wave vectors we have X 𝑦𝑥̂ 𝑬= 𝑬 (𝑆) + 𝑬 (𝑃) Perpendicular to plane of incidence In plane of incidence 𝑧𝑥̂ P. Piot, PHYS , NIU FA2018
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Equation for E field at the boundary
Expliciting in the E field gives Imposing the boundary condition, 𝐸 || continuous, yields in medium i in medium t P. Piot, PHYS , NIU FA2018
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Conditions for k and E (I)
Previous equation yields (equating the phase factors) where we assumed Using results in Law of reflection Snell’s law P. Piot, PHYS , NIU FA2018
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Conditions for k and E (II)
In addition the from the E-field equation (given that all phase are identical), we equate the components along each direction to obtain: We have four unknowns and two equation to supplemental equations can be derived from the B field associated with a plane wave see p.76 These equations are generally not convenient and instead one relies on the Fresnel’s coefficients. P. Piot, PHYS , NIU FA2018
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reflection laws using Fermat Principle
As a digression consider light can be model as a ray Fermat principle state “light travels between two points along the path that requires the least time, as compared to other nearby paths.” This is equivalent to the least-principle action in Classical Mechanics Consider the case of reflection, the total path from A to B is Minimize when or P. Piot, PHYS , NIU FA2018
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Snell’s laws using Fermat Principle
Same can be done for Snell’s law Minimum path given by which yields The point here is that although the EM description provide a physically more complete description of the light propagation, some features can be derive using simpler model hence the use of geometric (or ray) Optics. P. Piot, PHYS , NIU FA2018
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