The boundary conditions will determine the details of the refraction. Snells law is then used to characterize the refraction angle in terms of the complex wavenumber for both media. Snells law can be applied to all materials, in all phases of matter. Fresnel equations encyclopedia of optical engineering. Lecture 22 electromagnetic waves mit opencourseware. When a wave crosses a boundary between two isotropic media, the wave changes direction such that. Snells law can be derived using elementary calculus and trigonometry. To generate the law of refraction also known as snells law, we consider points pxi,1,rx,1 and sxt,1in the above diagram. Snells law is the generalization of the above in that it does not require the medium to be the same everywhere. Reflection and snells law electromagnetic geophysics. To determine the magnitudes, we need the boundary conditions at the. Consider three incident rays of light encountering an interface between two media.
Opticsfermats principle wikibooks, open books for an. As promised, lets do a couple of simple snells law examples. The derivation of snells law using fermats principle is straightforward. Is snells law incompatible with interface conditions.
The focus is both on snells law of refraction and, on the reflection principle under various conditions. The light is bending away from the normal as it enters a rarer material. This law was discovered in 1621 by the dutch astronomer and mathematician willebrord snell also called snellius. The issue is that snells law does not seem to hold when applied to the transmitted wave calculated using the electromagnetic interface conditions. The laws of electromagnetism, applied to this case, give the following boundary conditions. A ray of light beginning in the top medium travels through the interface into the bottom medium. Notice that this time, the angle of refraction is larger than the angle of incidence. I love that derivation, because we show that the boundary conditions that maxwells equations enforce lead directly to three things. The normal on the surface is used to gauge the angles that the refracted ray creates at the contact point.
The account of snells law went unpublished until its mention by christiaan. This appendix presents a derivation of the evanescent wave equation and discussion. Snells law, in optics, a relationship between the path taken by a ray of light in crossing the boundary or surface of separation between two contacting substances and the refractive index of each. Fermats principle states that light travels between two points along the path that requires the least time, as compared to other nearby paths. Another way to derive snells law involves an application of the general boundary conditions of maxwell equations for electromagnetic radiation. What is the direction of energy flux of the em wave. Boundary condition for the electric field at an interface. The law of refraction is also known as snells law, named for willobrord snell, who discovered the law in 1621. Boundary conditions 00 00 ee vv deriving boundary conditions slide 8 integral equations do not require boundary conditions as long as they do not contain derivatives. Boundary condition for the magnetic field at an interface. Suppose we have an interface at y0 between vacuum medium 1.
Here, is a unit vector pointing in the direction of wave propagation. Snells law like with reflection, refraction also involves the angles that the incident ray and the refracted ray make with the normal to the surface at the point of refraction. Snells law, also known as the law of refraction, is an equation that relates the angle of the incident light and the angle of the transmitted light at the interface of two different mediums. Suppose that the plane forms the boundary between two different dielectric media. In this lesson we are going to look at a derivation of snells law based on the principle of least time. In the diagram shown above, two mediums are juxtapositioned one below the other. In optics, the law is used in ray tracing to compute the angles of incidence or refraction, and in experimental optics to find the refractive index of a material. The perpendicular component of b is continuous across the boundary between the two media. And i know that i have a light ray, coming in with an incident. What is the basic principle behind the boundary conditions for em waves. From te analysis from phase matching from fermats principle of least time total internal reflection and fibers fios. Let medium 1, of refractive index, occupy the region, whilst medium 2, of refractive index, occupies the region. The law is also satisfied in metamaterials, which allow light to be bent backward at a negati. Fresnel equations consider reflection and transmission of light at dielectricdielectric boundary.
Snells law of refraction can be derived from fermats principle that light travels paths that minimize the time using simple calculus. The parallel component of e is continuous across the boundary between the two. Let us investigate what happens when an electromagnetic wave is incident on this boundary from medium 1. Reflection and refraction 17 be scattered away from its original direction figure 2. From fermats principle, one can derive a the law of reflection the angle of incidence is equal to the angle of reflection and b the law of refraction snells law. Calculate reflection and transmission coefficients, r and t, as a function of incident light polarisation and angle of incidence using em boundary conditions. Where i is the angle of incidence and r is the angle of refraction. Outline te and tm fields refraction and snells law. This problem is widely regarded as the founding problem of the calculus of variations. The elastic wave equationthe elastic wave equation elastic waves in infinite homogeneous isotropic media numerical simulations for simple sources plane wave propagation in infinite media frequency, wavenumber, wavelength conditions at material discontinuities snells law. Can the snells law be applied in the case where a ray of light is incident with some angle on the rectangular slab whose refractive index changes with depth of the rectangular slab and refractive index has a linear relation with depth of the slab. For the love of physics walter lewin may 16, 2011 duration. Refraction and snells law reading shen and kong ch. Maxwells equations and boundary conditions at interfaces.
The position of the entire free surface not just the exit point is unknown a priori. T he same approach, however, can be applied in an analogous manner to both shear and converted waves slawinski, 1996. Snells law also known as snelldescartes law and the law of refraction is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water, glass, or air in optics, the law is used in ray tracing to compute the angles of. The wavenumbers will be related to the frequency differently according to the speed of light in the materials. Since snells law only involves sines i wonder whether this minimum problem has a simple geometric solution. For example, suppose that medium 1 in figure 2 is air and medium 2 is water. Using the chain rule, the lefthand side of equation 8 can be rewritten to the form of explicit secondrank di erential equation. Questions you should be able to answer by the end of todays lecture. But, suppose that the path \qo\ through which the light ray traversed is a different medium than that of the surface and below the surface. Its a really cool argument that comes down to this. Proof of snells law using maxwells equations physics. Pdf generalized form of snells law in anisotropic media.
There are further conditions on the amplitudes e0 and b0 from the other maxwell equations. I have labeled these different media medium 1 and medium 2 in figure 2. For this reason, we will derive our boundary conditions using maxwells equations in integral form. Fermats principle and the laws of reflection and refraction. If sinr exceeds 1 as given by this equation, a head wave will be generated. A ray is a line that has an origin but does not have an end. Appendices appendix a derivation of the evanescent wave. The governing equation for sound in a honmogeneous. The fresnel equations and brewsters law equipment optical bench pivot, two 1 meter optical benches, green laser at 543. The scattered light goes off in many different directions, and may be scattered again and again before it is finally absorbed. Derivation from conservation of energy and momentum.