Introduction to Polarization

Transcription

1 Phone: Ext 659, Fall/007 Introduction to Polarization Text Book: A Yariv and P Yeh, Photonics, Oxford (007) 1.6 Polarization States and Representations (Stokes Parameters and Poincare Sphere) Here we introduce the concept of polarization by considering a beam of monochromatic plane waves propagating in an isotropic and homogeneous medium. The beam can be represented by its electric field E r, t, which can be written E = A cos ωt k r where ω is the angular frequency, k is the wavevector, and A is constant vector representing the amplitude. The magnitude of the wavevector k is related to the frequency by the following equation: k = n ω c = n π λ k E = 0 E = Aexp i ωt k r For the purpose of describing various representations of the polarization states, we consider propagation along the z axis. Because a transverse wave, the electric field vector must lie in the xy plane. The two mutually independent components of the electric field vector can be written E x = A x cos ωt kz + δ x E y = A y cos ωt kz + δ y The time evolution of the electric field vector at the origin (z=0) is shown E x = A x cos ωt + δ x E y = A y cos ωt + δ y We defined a relative phase as δ = δ y δ x, and δ is limited to the region π < δ π Linear Polarization state A beam of light is said to be linear polarization state if the electric field vector vibrates in a constant direction (in the xy plane). This occurs when the two 1

2 Phone: Ext 659, Fall/007 components of oscillator are in phase (δ = δ y δ x = 0) or π out of phase (δ = δ y δ x = π); that is δ = δ y δ x = 0 or π. In this case, the electric field vector vibrates sinusoidally along a constant direction in the xy plane defined by the ratio of the two components, E y E x = A y A x or A y A x At a fixed time (t=0), the components of the electric field vector can be written E x = A x cos kz + δ x and E y = A y cos kz + δ y with δ = δ y δ x = 0 or π. Circular Polarization States The other important case is circular polarization state. A beam of light is said to be circularly polarized if the electric field vector undergoes uniform rotation in xy plane. This occurs when A y = A x and δ = δ y δ x = ± π. In this textbook, the beam of light is right-handed circularly polarized when δ = π, which corresponds to a counterclockwise rotation of the electric field in xy plane. And left-handed circularly polarized when δ = π, which corresponds to a clockwise rotation of the electric field in xy plane. Elliptic Polarization States A beam of light is said to be elliptically polarized if the curve traced by the end point of the electric field vector is ellipse. This is most general case of a polarized light. The equation of the ellipse can be obtained as E x A x + E y A y cos δ A x A y E x E y = sin δ (1.6-1) (1.6-1) is an equation of a conic. It is obvious this conic section is confined in a rectangular region with side parallel to the coordinate axes and whose lengths are A x anda y. Let x and y be the new set of axes along the principal axes of the ellipse. Here the equation of the ellipse is this new coordinate system becomes E x + E y a b = 1

3 Phone: Ext 659, Fall/007 Where a and b are the lengths of the principal semiaxes, and E x and E y are the components of the electric field vector in this principal coordinate system. Let φ be the angle between the x axis and the x axis. Then the lengths of the principal axes are given by a = A x cos ϕ + A y sin ϕ + A x A y cos δ cos ϕ sin ϕ b = A x sin ϕ + A y cos ϕ + A x A y cos δ cos ϕ sin ϕ The angle φ can be expressed in terms of Ax, Ay, and cos δ as tan ϕ = A x A y A x A y If φ is a solution of the equation, φ+π/ is also a solution. The sense of revolution of an elliptical polarization is determined by the sign of sin δ. The end point of the electric vector will revolve in a clockwise direction if sin δ>0 and in a counterclockwise direction if sin δ<0 cos δ 3

4 Phone: Ext 659, Fall/007 The ellipticity of a polarization ellipse is defined as e = ± b. a and b are the a length of the principal semiaxes. The ellipticity is taken as positive when the rotation of the electric field vector is right-handed and negative otherwise. With this definition, e=±1 for circularly polarized light. Complex-Number Representation X = e iδ tan ψ = A y A x e i δ y δ x The inclination angle φ and the ellipticity angle θ (θ tan -1 e) of the polarization ellipse corresponding to a given complex number χ are given by tan ϕ = Re χ Im χ 1 χ = tan ψ cos δ and sin θ = 1+ χ = sin ψ sin δ Jones Vector Representation: 4

5 Phone: Ext 659, Fall/007 The Jones vector is used for description of the polarization state of a plane wave. Here, the plane wave is expressed in terms of its complex amplitudes as a column vector J = A xe iδ x A y e iδ y J J = 1 A beam of linearly polarized light with the electric vector oscillating along a given direction can be represented by the Jones vector cos ψ sin ψ. ψ is the azimuth angle of the oscillation direction with respect to the x axis. The orthogonal state of polarization can be obtained by ψ + π for ψ, leading to a Jones vector sin ψ cos ψ The special case, when ψ = 0, represents linearly polarized waves whose electric filed vector oscillates along the coordinate axes. The Jones vector are given by x = 1 0 and y = 0 1. Jones vectors for the right- and left-handed circularly polarized light wave are given by R = 1 1 i and L = 1 mutually orthogonal in the sense that R L = 0 1 i. These two states of circular polarizations are Any polarization state can be represented as a superposition of two mutually orthogonal polarization states x and y, or R and L In particularly, R = 1 x iy and L = 1 x + iy x = 1 R + L and y = i R L A general elliptic polarization state can be represented by the following Jones vector: J ψ, δ = cos ψ e iδ sin ψ. 5

6 Phone: Ext 659, Fall/007 Stokes Parameters and Partially Polarized Light: To describe the polarization state of this type of radiation, we introduce the following time-averaged quantities: S 0 = A x + A y S 1 = A x A y S = A x A y cosδ S 3 = A x A y sinδ Where A x anda y and the relative phase δ are assumed to be time dependent, and the double brackets denote averages performed over a time interval τ D that is the charactersitci time constant of the detection process. These four quantities are known as the stokes parameters of a quasi-monochromatic plane wave. Four quantities have the same dimension of intensity. It can be shown that the stokes parameters satisfy the relation S 1 + S + S 3 S 0. The equal sign holds only for polarized waves. If the beam is totally unpolarized, S 1 = S = S 3 = 0, and if the beam is 6

7 Phone: Ext 659, Fall/007 completely polarized, S 1 + S + S 3 = 1. The degree of polarization is defined as γ = S 1 + S + S 3 1 S 0 The Stokes parameters for a polarized light with a complex representation X = e iδ tan ψ are given by S 0 = 1 S 1 = cos ψ S = sin ψcosδ S 3 = cos ψsinδ According to our convention, a positive S 3 corresponds to left-hand elliptical polarization (sinδ > 0, clockwise revolution) Poincare Sphere For polarized light, the Stokes parameters S 1, S, and S 3 can also be employed to represent the polarization states. Since S 0 =1, all points with coordinate (S 1, S, S 3 ) are confined on the surface of a unit sphere in 3-D space. This sphere is known as the Poincare Sphere. Consider the two different points on the Poincare sphere. Each point represents a polarization state. Let the Stokes vector be written S a = 1, S a1, S a, S a3 and S b = 1, S b1, S b, S b3 Using (1.6-33), it can easily be shown that S a S b = J a J b (1.6-39) Where J a and J b are the corresponding Jones Vectors. For polarized light, the first component of the stokes vectors is 1, so it is convention to define three-dimension unit vectors, consisting of the three components S 1, S, and S 3 of the Stokes vectors as S a = S a1, S a, S a3 and S b = S b1, S b, S b3. These three-component unit vectors are real vectors in Poincare sphere. The tips of these unit vectors correspond to points on the Poincare sphere. With this definition, (1.6-39) can be written J a J b = 1 S a S b = S a S b For a pair of antipodal points, S a S b = 1, the two polarization states form an orthogonal pair. 7

8 Phone: Ext 659, Fall/ Electromagnetic Propagation in Anisotropic Media (Crystals) In anisotropic media, the propagation of EM radiation is determined by the dielectric tensor ε ij that lines the displacement vector and the electric field vector. D i = ε ij E j In nonmagnetic and transparent materials, the tensor is real and symmetric: ε ij = ε ji Because of its real and symmetric nature, it is always possible to find three mutually orthogonal axes in such a way that the off-diagonal elements vanish, leaving ε = ε 0 n x n y n z = ε x ε y 0 (1.7-3) 0 0 ε z where ε x, ε y, and ε z are the principal dielectric constants, and n x, n y, and n z 8

9 Phone: Ext 659, Fall/007 the principal indices of refraction. Plane Wave in Homogeneous Media and Normal Surface To study such propagation along a general direction, we assume a monochromatic plane wave with an electric field vector E exp i ωt k r, and a magnetic field vector H exp i ωt k r. k is the wavevector k = ω/c ns, with s as a unit vector in the direction of propagation. Substitute E and H into curl Maxwell s equations gives k E = ωμh and k E = ωμe = ωd By eliminating H, we obtain k k E + ω μεe = 0 (1.7-8) The equation will now be used to solve for the eigenvectors E and the corresponding eigenvalues n. In the principal coordinate system, the dielectric tensor is given by (1.7-3). And (1.7-8) can be written ω με x k y k z k x k y k x k z k y k x ω με y k x k z k y k z k z k x k z k y ω με z k x k y E x E y = 0 (1.7-9) E z For nontrivial solutions to exist, the determinant of the matrix must vanish. This leads to a relation between ω and k: det ω με x k y k z k x k y k x k z k y k x ω με y k x k z k y k z k z k x k z k y ω με z k x k y = 0 (1.7-10) At a given frequency ω, this equation represents a 3-Dimesion surface in k space (momentum space). This surface is the normal surfaces and consists of two shells. These two sells have four points. The two lines that go through the origin and these points are known as the optical axes. 9

10 Phone: Ext 659, Fall/007 The directions of the electric field vector associated with these propagations can be obtained and are given by k x k ω με x k y k ω με y k z k ω με z (1.7-11) By using the relation k = ω/c ns for the plane wave, (1.7-10) and (1.7-11) can be written as s x n n + s y x n n + s z y n n = 1 z n (1.7-1) and s x n n x s y n n y s z n n z (1.7-13) Orthogonality of Normal Modes (Eigenmodes) It can be shown that D, E, and s all lie in the same plane. These field vectors satisfy the following relations: D 1 D = 0 D 1 E = 0 D E 1 = 0 s D 1 = s D = 0 The electric field vectors E 1 and E are not orthogonal. The general orthogonality relation of the Eigenmodes of propagation is often written s E 1 H = 0 10

11 Phone: Ext 659, Fall/007 Classification of Media n x, n y, n z. The normal surface is uniquely determined by the principal indices of refraction In the general case when the three principal indices n x, n y, n z are all different, there are two optical axes. In this case, the medium is said to be biaxial. In uniaxial medium, two of the principal indices are equal, and then the equation for the normal surface can be factored according to k x + k y n e Here, n o = ε x ε 0 = ε y ε 0 and n e = ε z ε 0 + k z n ω k o c n ω o c = 0 The Index Ellipsoid The surface of constant energy density U e in D space can be written D x + D y + D z = U ε x ε y ε e z Where ε x, ε y, and ε z are the principal dielectric constants. If replace D/ U e by r and use the principal refractive indices n i = ε i written as x n x + y n y + z n z = 1 ε 0 i = x, y, z, this equation can be This is the equation of a general ellipsoid with major axes parallel to the principal axes of the crystal, whose respective lengths are n x, n y, n z. The ellipsoid is known as the index ellipsoid. The index ellipsoid is used mainly to find the D vectors and the two corresponding indices of refraction of the normal modes of the propagation along a given direction of propagation s in a crystal. 11

12 Phone: Ext 659, Fall/ Plane Wave in Uniaxially Anisotropic Media-Phase Retardation Many optical electronic material are optical uniaxial. A planar nematic is a good example of a homogenous uniaxial liquid crystal. For discussing the propagation of optical waves in uniaxial anisotropic media, we rewrite the moral surface of a uniaxial medium as k x + k y n e + k z n ω k o c n ω o c = 0 We note that normal surface consist of two parts. The sphere gives the relation between w and k of the ordinary (O) wave. The ellipsoid of revolution gives the similar relation for the extraordinary (E) wave. These two surfaces touch at two points on the z axis. The eigen refractive indices associated with these two modes of propagation are given by O wave: n = n o and E wave: 1 = cos θ n n + sin θ o n with θ is the angle between the e direction of propagation and the c axis. For propagation along the optic axis, the refractive index is n 0. The electric field vector of the O wave can be obtained from (1.7-9). By using ε x = ε y = ε 0 n o, ε z = ε 0 n e and k o = ω/c n o s, (1.7-9) can be written s x s x s y s x s z s y s x s y s y s z s z s x s z s y n e n o s x + s y E x E y E z = 0 A simple inspection of this equation yields the following direction of polarization: 1

13 Phone: Ext 659, Fall/007 O wave: E = s y s x 0 by The electric field vector of the E wave can be obtained from (1.7-13) and is given E wave: E = s x n n o s y n n o s z n n e where n e is given by (1.8-3) and k e = ω/c n e s. The displacement vectors D of the normal modes are exactly perpendicular to the wavevectors k o and k e and be written O wave: D o = k o c k o c E wave: D e = D o k e D o k e where c is a unit vector parallel to the c axis of the crystal. Let (θ, φ) be the angle of propagation in spherical coordinates. The unit vector s can be written s = sin θ cos ϕ sin θ sin ϕ cos θ Using (1.8-9), the normal modes for E can be written O wave: E o = sin ϕ cos ϕ 0 E wave: E e = n e cos θ cos ϕ n e cos θ sin ϕ n 0 sin θ The normal modes for D can be written: 13

14 Phone: Ext 659, Fall/007 O wave: D o = E wave: D e = The two D vectors and s are mutually orthogonal. sin ϕ cos ϕ 0 cos θ cos ϕ cos θ sin ϕ sin θ If inside the uniaxial medium a polarized light is generated that is to propagate along a direction s, the displacement vector of this light can always be written as a linear combination of these two normal modes; that is D = C o D o exp ik o r + C e D e exp ik e r with C o and C e are constant and k o and k e are the wavevector that are different. As the light propagates inside the medium, a phase retardation between these two components is built up due to the difference in their phase velocities and leads to a new polarization state. Birefringent plates can be used to alter the polarization state of light. For a parallel plate with thickness d, the phase retardation can be written Γ = k ez kozd with kez and koz are the z components of the wavevectors and the z axis is perpendicular to the surface of the plates. Optical Rotatory Power and Faraday Rotation The optical activity arises from circular double refraction, in which the eigenwaves of propagation are right- and left- circularly polarized waves. Let n r and n l be the refractive indices associated with these two waves, 14

15 Phone: Ext 659, Fall/007 assume that the waves are propagating in the +z direction. It can be shown that the specific rotatory power is given by ρ = π λ n l n r Faraday Rotation The specific rotation of a Faraday cell is often written: ρ = VB with B is the component of magnetic field along the direction of propagation and V is a constant known as the Verdet constant. Faraday Isolator and Optical Circulator Faraday Effect plays an important role in nonreciprocal optical devices such as isolators and circulators. 15

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