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**Polarization of EM waves**

The polarization of a uniform plane wave refers to the time-varying behaviour of the electric field vector at fixed point in space. Consider, for example, a uniform plane wave travelling in the z-direction. With the E and H vectors lying in the x-y plane. Case 1: If Ey=0, and only Ex is present: The wave is said to be polarized in the x direction. If Ex=0, and only Ey is present: The wave is said to be polarized in the y direction.

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**If both Ex and Ey are present, and are in phase (=0, 2, 4**

If both Ex and Ey are present, and are in phase (=0, 2, 4....) the resultant electric vector has a direction, dependent on the relative magnitude of Ex and Ey. The resultant vector will make the angle tan-1(Ey/Ex). In all the above cases the wave is said to be linearly polarized because the resultant vector is constant with time. Case 2: If both Ex and Ey are not in phase (=(2n+1)/2, n=0, 1, ), i.e. their maximum values (of amplitude) reach at different time, the direction of resultant electric vector will vary.

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The locus of the end point of the resultant E will be an ellipse and the wave is said to be elliptically polarized. Case: 3 When Ex and Ey have equal magnitudes and not in phase i.e.(=(2n+1)/2, n=0, 1, ), the locus of the resultant E will be an circle and the wave is said to be circularly polarized.

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**Equation (1) can be written as:**

Let the two plane polarized EM waves may superimpose. One has Electric vector, Ex in X direction and other has electric vector, Ey in Y direction, and both are propagating in Z direction. Where a and b are the amplitude of the waves and is the phase difference between them. Equation (1) can be written as:

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From 2nd equation: From above equation we can find: Put these values in equation 3: or or

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or or This equation in general represents an ellipse. Therefore in this case the resultant will be elliptically polarised light. But actual nature of resultant wave can be given by following cases: Case I: n: integer then

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So equation 4 will be: or or or Which is the equation of a straight line. Thus emergent ray in this case will be plane polarised light.

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Case II: then So equation 4 will be: Which is the equation of an ellipse. Thus emergent ray in this case will be elliptically polarised light.

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Case III: with a=b So equation 4 will be: Which is the equation of a circle of radius a. Thus emergent ray in this case will be circularly polarised light.

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