Polarization of Uniform Plane Waves

Polarization of Uniform Plane Waves

• The polarization of uniform plane waves is defined as time varying behaviour of the electric field intensity vector  at some fixed point in space, along the direction of propagation.

• Consider a uniform plane wave travelling in positive z-direction. Then the field vectors  and  lie in x-y plane, which is perpendicular to the direction of propagation of a uniform plane wave. Being an electromagnetic (EM) wave, as wave travels in a space, both the fields undergo same variations with respect to time.

• There are different types of polarization of a uniform plane wave as given below.

1) Linear polarization

2) Elliptical polarization

3) Circular polarization

• In other words, the polarization is nothing but a way in which the magnitude and direction of the electric field varies. These variations of the field are observed in the direction of propagation of a wave. So if the variation of the electric field  is observed on a plane perpendicular to the direction of propagation of wave, we get  along straight line or circle or ellipse.

• For the understanding of basic concept of polarization of a uniform plane wave, let us assume that  is the resultant of  are directed along x-direction and y-direction respectively. Also assume that the variations of  be sinusoidal similar to . To get different types of the polarization for , consider different conditions for .

 

1. Linear Polarization

• Consider that the electric field  has only x-component and y-component of  is zero. Then looking from the direction of propagation, the wave is said to be linearly polarized in x-direction. Similarly if only the y-component in  is present and x-component of  is zero then the wave is said to be linearly polarized in y-direction.

• Let us assume that both the components of  are present denoted by . Both these components are in phase having different  amplitudes. As  are in phase they will have their amplitudes reaching maximum or minimum value simultaneously. Also if the amplitude of  increases or decreases the amplitude of  also increases or decreases. In other words, at any point along positive z-axis the ratio of amplitudes of both the components is constant as both of them are in phase having same wavelength.

• The electric field is the resultant of  and the direction of it depends on the relative magnitude of . Thus the angle made by  with x-axis is given by,

θ = tan-1 E_y / E_x

• Where E_x and E_y are the magnitudes of  respectively.

• This angle is constant with respect to time. In other words, the resultant vector  is oriented in a direction which is constant with time, thus the wave is said to be linearly polarized as shown in the Fig. 10.9.1.

• When both the components have same amplitudes we get a polarization of  as linear polarization with a constant angle of 45°.

• From Fig. 10.9.1 it is clear that when Ex increases or decreases, Ey also increases and decreases as both are in phase. And the resultant  changes along the straight line at an angle θ° with respect to x-axis.

• Thus when  components are in phase with either equal or unequal amplitudes, for a uniform plane wave travelling in z-direction, the polarization is linear.

 

2. Elliptical Polarization

• Consider that the electric field  has both the components which are not having same amplitudes and are not in phase. As the wave propagates,  will have maximum and minimum amplitudes at different instants of time depending on the relative amplitudes of  at any instant of time. In other words, the direction of the resultant field  varies with time. If the locus of the end points of  is traced, it is observed that  moves elliptically. Then such a wave is said to be elliptically polarized as shown in the Fig. 10.9.2.

• When the amplitudes of  are different and the phase difference between the two is other than 90°, then the axes of the ellipse are inclined at an angle 0 with the co-ordinate axes.

• When this phase difference between  is exactly 90°, then the axes of the ellipse lie along the co-ordinate axes.

• Let us consider that the amplitudes of  components are unequal. Let the phase difference between the two components be 90° exactly. The maximum amplitude of will occur at different instants of time. Thus the direction of   varies with time, along an ellipse as shown in the Fig. 10.9.3. Refer Fig. 10.9.3

With 90° phase difference between ;

i) The major axis of an ellipse lies along y-axis if the amplitude of  is greater than that of the component and

ii) The major axis of an ellipse lies along x-axis, if the amplitude ofis greater than that of 

• Thus when the components  of unequal amplitudes have a constant, non-zero phase difference between two, for a uniform plane wave z-direction, the polarization is travelling in elliptical.

 

3. Circlar Polarization

• Let us consider that  has two components,  , of equal amplitude but the phase difference between them is exactly 90°. At any instant of time, when the amplitude of any one of the component is maximum, then the amplitude of the remaining component becomes zero. Also when any component gradually increases or decreases; the other component gradually decreases or increases respectively. Thus at any instant of time the magnitude of the resultant vector   is constant. But the direction of the resultant vector   changes with respect to the time as the angle depends on the relative amplitudes of the two components at every instant. If   is projected on a plane perpendicular to the direction of propagation, then the locus of all such points is a circle with the centre on z-axis. In one wavelength span, the resultant vector   completes one cycle of rotation. Then such a wave is said to be circularly polarized as shown in the Fig. 10.9.4 Refer Fig. 10.9.4.

• If   is having components in both x and y-directions, we can write,

• Where δ is the phase shift between two components.

Consider that field   is observed at z = 0. Then

• Graphically it is represented as shown in the Fig. 10.9.5.

• From the Fig. 10.9.5 it is clear that the resultant vector  rotates in clockwise direction, such that the locus of all such points represents a circle. As per IEEE definitions, this type of polarization is called left circular polarization.

Graphically resultant  can be represented as shown in the Fig. 10.9.6.

• From the Fig. 10.9.6 it is clear that the resultant vector  rotates in anticlockwise direction, such that the locus of all such points represents a circle. As per IEEE definitions, this type of polarization is called right circular polarization.

• Any electric wave has two components ;

i) Forward travelling component E_m sin (ωt - βz)

ii) Backward travelling component E_m sin (ωt + β z)

• Note that the definitions of right and left circular polarization are obtained for the forward travelling wave. For backward travelling wave the definitions of right and left circular polarization are exactly opposite to the previous one.

• Thus when two components   of equal amplitudes have a constant, non-zero phase difference of 90° between them, for a uniform plane wave, the polarization is circular.

 

4. Conditions for the Polarization of a Sinusoidal Wave

• Consider that the electric field  of a uniform plane wave travelling in z-direction is expressed as

• This can be expressed in the time varying form as

• As the wave propagates in z-direction,  must lie in a plane perpendicular to the direction of propagation i.e. x-y plane. Thus is a resultant of two components  along x-axis and y-axis respectively. Let E₁ and E₂ be the amplitudes of  respectively. Assume that both the variations are sinusoidal. Let δ be the phase difference between the two components.

• The two components can be expressed in phasor form as,

• As the electric field vector E is the resultant of Ex and Ey, we can write,

• Equation (10.9.5) is true for all values of z. At z = 0, the equation (10.9.5) can be rewritten as,

• In general, the electric field vector can be represented with its two components as,

• Equating right hand side of equation (10.9.7) with the real part of the equation (10.9.6),

• Above equation is the equation for polarization of sinusoidal wave. By applying different conditions to equation for the polarization of sinusoidal wave, we get different types of the polarization.

Condition 1 :  components are in phase i.e. δ = 0.

• Substituting this condition in equation (10.9.14) we can write,

• For a given wave, the amplitudes of  remain constant i.e. Ei and E2 are constant. Thus the ratio (E₁ / E₂) is also constant. Then the equation (10.9.15) is similar to the equation of a straight line passing through origin, i.e. y = mx. Then the wave is said to be linearly polarized wave.

Condition 2 :  components of unequal amplitudes with a phase difference δ ≠ 0. Let us assume that δ = π/ 2^c.

• Applying conditions to equation (10.9.14), we can write,

We know that,

sin² θ  = 1/2 (l - cos² θ), hence we get sin²1 (π/ 2) = 1

• Then equation becomes

(E_x / E₁)² + (E_y / E₂)²  = 1 ... (10.9.16)

• Equation (10.9.16) represents equations for an ellipse. Thus, the wave, with the components specified in the condition, is said to be elliptically polarized.

Condition 3 :  components of equal amplitude with the phase difference between two as

δ = π^c / 2

• Let the amplitude of  be equal to E₀.

• Applying conditions to equation (10.9.14), we can write,

• Equation (10.9.17) represents equation of a circle similar to x² + y² = a². Thus the wave, with the components specified in the condition is said to be circularly polarized.

 

Ex. 10.9.1 The electric field of a uniform plane wave is given by

find the polarization of the wave.

Sol. : The electric field of an uniform plane wave is given by,

• From above equation it is clear that the magnitudes of x and y components of   are equal. (E_x = E_y = 10). At the same time the phase difference between these two components is 90s. Hence the wave is circularly polarized.

Review Questions

1. What is polarization of uniform plane waves ? Write notes on

i) Linear polarization,

ii) Elliptical polarization

iii) Circular polarization.

2. Derive the condition for polarization of a sinusoidal wave.