Snell’s law shows the relationship between angles of incidence and refraction, and the refractive index of each medium. The refractive index refers to the extent to which a medium can increase or decrease the speed of light. The image below shows how refraction works.
According to Snell’s law,
n1Sinθ1= n2Sinθ2
where n1 is the refractive index of medium 1 and n2 is the refractive index of medium 2. θ1 is the angle of incidence and θ2 is the angle of refraction.
All these angles lie on the same surface. The image below shows the laws of reflection and how it works.
According to Huygen, every point on a wavefront is a source of secondary wavelets that spread out in a forward direction at the speed of the wave itself. The new wavefront is the sum of all the secondary wavelets.
According to Huygen’s Principle, a plane light wave moves through free space at the speed of light. The light waves associated with this wavefront also move in a straight line as shown in the image below.
Figure info: Reflection of plane wave at plane surface using wavefront
Consider the image shown below, where plane wavefront AB, on the surface XY, where AB is the incident ray reflecting on the surface at point A.
PA and QBC are perpendiculars drawn to points A and B respectively, making them incident rays. AN is the normal line drawn on the surface. According to Huygen’s Principle, A and B will act as a source of multiple secondary wavelets. Consider A and B as sources for secondary wavelets. The time taken for the secondary wavelets to travel the distance BC from point B is the same time taken for secondary wavelets from A to travel the distance BC after reflection. This makes CD the reflected plane wave and AD the reflected ray.
Now, we can see the incident wavefront AB, the reflected ray CD and the surface XY are on the same plane. Refer to the image below to see how Huygen’s Principle is used to explain Laws of Reflection.
Angle of incidence i = ∠PAN = 90° – ∠NAB = ∠BAC
Angle of reflection r = ∠NAD = 90° – ∠DAC = ∠DCA
In right angled triangles ABC and ADC
∠B = ∠D = 90°
BC = AD and AC is common
The two triangles are congruent
∠BAC = ∠DCA
i.e. i = r
Thus the angle of incidence is equal to the angle of reflection.
Consider a plane wavefront XY that separates the rare medium ( μ1) from the denser medium (μ2). Here, the velocities of the incident ray and refracted ray of the rare medium and denser medium can be v1 and v2 respectively. The velocity of the waves depends upon the medium. Following Huygen’s Principle, A and B form the source of secondary spherical wavelets. Here, ‘t’ can be the time taken from B to reach B’.
So in medium 1, BB’ = v1t.
Simultaneously, the secondary wavelets originating from point A will travel the same distance in the denser medium to A’. Therefore,
AA’= v2t in medium 2.
Now A’B’ are the secondary wavefronts at time t.
Sin i / Sin r= (BB’/AB’) / (AA’/AB’)
= BB’/AA’
= v1t/v2t
= v1/v2
= μ which is a constant.
μ is the reflective index of the medium. The refractive index is the ratio of the velocity of light in a vacuum to the velocity of light in another medium. Thus, this proves Snell’s law of refraction using plane waves and Huygen’s Principle.
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