The Hong Kong Polytechnic University Polarization Consider the

The Hong Kong Polytechnic University Polarization: Consider the superposition of two plane polarized waves: If =2 m , m being an integer, If =(2 m+1) , m being an integer, Optics II----by Dr. H. Huang, Department of Applied Physics 1

The Hong Kong Polytechnic University Right Circularly Polarized Light: If =(2 m-1/2) , m being an integer, Polarization Left Circularly Polarized Light: If =(2 m+1/2) , m being an integer, Right and left circular light can be written as, Their superposition becomes A plane polarized wave can be synthesized from two oppositely polarized circular waves. Optics II----by Dr. H. Huang, Department of Applied Physics 2

The Hong Kong Polytechnic University Polarization Elliptically Polarized Light: Suppose and =(2 m+1/2) , m being an integer. State of Polarization: P-state R-state and L-state E-state Natural Light: Natural light is randomly polarized. We can mathematically represent natural light in terms of two arbitrary, incoherent, orthogonal, linearly polarized waves of equal amplitude (i. e. , waves for which the relative phase difference varies rapidly and randomly). Optics II----by Dr. H. Huang, Department of Applied Physics 3

The Hong Kong Polytechnic University Polarization Example: A wave has the components x=E 1 cos(kz- t) and y=-E 1 cos(kz- t). What is its state of polarization? Polarizers: a device to generate polarized light out of natural one. It can also be used as an analyzer to allow all E-vibrations parallel to the transmission axis to pass. Malus’ Law: When a natural light passes though an ideal polarizer, its intensity is reduced by half. Optics II----by Dr. H. Huang, Department of Applied Physics 4

The Hong Kong Polytechnic University Polarization Dichorism: selective absorption of one of the two orthogonal P-state in incident natural light. wire-grid polarizer dichroic crystal Commercial Polaroid H-Sheet: • It’s a dichroic sheet polarizer. • An ideal H-sheet would transmit 50% of the incident natural light and is designated HN-50. • In practice, due to loss, the H-sheet might be labeled HN-46, HN-38, HN-32, and HN-22 with the number indicating the percentage of natural light transmitted through the H-sheet. Optics II----by Dr. H. Huang, Department of Applied Physics 5

The Hong Kong Polytechnic University Polarization Example: Natural light of intensity Ii is incident on three HN-32 sheets of Polaroid with their transmission axes parallel. What is the intensity of the emergent light? Example: Suppose third Polaroid in the last question is rotated through 45. What is now the intensity transmitted? Example: Two sheets of HN-38 Polaroid are held in contact in the familiar crossed position. Let I 1 be the intensity emerging from sheet 1 natural light of intensity Ii is incident upon it. The intensity emerging from sheet 2 must be I 2=0. Now insert a third sheet of HN-32 between them with its transmission axis at 45 to the other sheets’ transmission axes. What is I 2 now? Optics II----by Dr. H. Huang, Department of Applied Physics 6

The Hong Kong Polytechnic University Polarization Birefringence: A material which displays two different speeds of propagation in fixed and orthogonal directions, and therefore displays two refractive indices, is known as birefringent. Distinction: A dichroic material absorbs one of the orthogonal P-states is dichroic while in birefringent material we usually neglect the absorption. Atomic structure of a Ca. CO 3 tetrahedron. Rhombohedron of calcite. The optic axis passes symmetrically through a blunt corner where three face angles equal 102. is called birefringence. uniaxial positive uniaxial negative Optics II----by Dr. H. Huang, Department of Applied Physics 7

The Hong Kong Polytechnic University Double Refraction: Polarization A narrow beam of natural light incident normally to a cleavage plane of a calcite crystal emerges as two parallel beams displaced laterally. Creation of an elliptical Huygens’ wavelet by extraordinary ray. The material is uniaxial negative. Ray direction S for the extraordinary ray in birefringence material. Optics II----by Dr. H. Huang, Department of Applied Physics 8

The Hong Kong Polytechnic University Polarization Example: A calcite plate is cut as shown in figure with the optic axis perpendicular to the plane of the paper. A ray of natural light, =589. 3 nm, is incident at 30 to the normal. The plane of the paper is the plane of incidence. Find the angle between the rays inside the plate. Optics II----by Dr. H. Huang, Department of Applied Physics 9

The Hong Kong Polytechnic University Polarization Various Types of Plane Polarizers: Glan-Air prism Nicol prism Optics II----by Dr. H. Huang, Department of Applied Physics 10

The Hong Kong Polytechnic University Polarization Example: A 50 calcite prism is cut with its optic axis as shown in the figure. Sodium light is used in a spectrometer experiment to find no and ne. Two images of the slit are seen and minimum deviation is measured for each. Find no and ne if the angles of minimum deviation are 27. 83 and 38. 99. Explain how you would decide which image was formed by the o-rays and which was formed by the e-rays. Since calcite is uniaxial negative, therefore ne=1. 4864 and no=1. 6584 Optics II----by Dr. H. Huang, Department of Applied Physics 11

The Hong Kong Polytechnic University Polarization Example: A quartz Wollaston beam-splitting polarizer is used with a normally incident parallel beam of sodium light for which ne=1. 5534 and no=1. 5443. If the wedge angle is 45 , find the angular separation of the emergent e- and o-rays. Optics II----by Dr. H. Huang, Department of Applied Physics 12

The Hong Kong Polytechnic University Polarization Scattering: The displacement of an electron oscillating harmonically under an external field is where 0 is the natural frequency (resonant frequency) of the oscillation of the bound electron. At resonant frequency, strong absorption occurs. At non-resonant frequencies the absorption of the wave packet and its subsequent emission is known as scattering. Rayleigh Scattering: Scattering centers have dimensions smaller than the wavelength. The radiated power is inversely proportional to the fourth power of the wavelength of the incident radiation. Polarization due to scattering: Optics II----by Dr. H. Huang, Department of Applied Physics 13

The Hong Kong Polytechnic University Polarization by Reflection: At Brewster’s angle ip, the reflected light becomes plane polarized perpendicular to the plane of incidence. pile-of-plates polarizer Brewster window Degree of polarization: Optics II----by Dr. H. Huang, Department of Applied Physics 14

The Hong Kong Polytechnic University Polarization Example: The figure shows a ray of monochromatic light incident at an angle ip on extra-dense flint glass for which ng=1. 653. Show that the angle between the transmitted and reflected rays is generally equal to 90 , and find ip and i in this particular case. Example: Natural light is incident on a plane air/glass boundary. Suppose the angle of incidence is about 55 and the refractive index of glass is 1. 5. Calculate the degree of polarization. Optics II----by Dr. H. Huang, Department of Applied Physics 15

The Hong Kong Polytechnic University Polarization Retarders: Retarders are devices that cause one orthogonal P-state component to lag behind the other on emerging from the retarders. The path difference is Full wave plate: Half-wave plate: Quart-wave plate: The component E// and E that travels faster defines the fast axis of the plate. Optics II----by Dr. H. Huang, Department of Applied Physics 16

The Hong Kong Polytechnic University Polarization Compensator: ―A device that allows a continuous adjustment of the relative phase shift, the retardance. Soleil-Babinet compensator. (Left) Zero retardation. (Right) Maximum retardation. A circular light can be changed into Pstate with a quarter-wave plate. The handiness of circular light can be checked by this method. Optics II----by Dr. H. Huang, Department of Applied Physics 17

The Hong Kong Polytechnic University Polarization Example: A beam of left circular light meets a quarter-wave plate with its fast axis vertical. What is the state of the emergent light? x leads y by /2. On passing through the wave plate, the y component travels faster than the x one. It emerges with a phase advanced by /4 and the two components are now in phase. The resultant wave is a P-state, making an angle =45 with the x-axis. Optics II----by Dr. H. Huang, Department of Applied Physics 18

The Hong Kong Polytechnic University Polarization Example: A quarter-wave plate made from calcite, ne=1. 4864 and no=1. 6584, is placed in a beam of normally incident light, =656 nm, plane polarized at 45 to the x-axis. Right circular light emerges. Find the minimum thickness of the plate and the orientation of the optic axis. For minimum thickness, m=0, therefore, t=953. 49 nm. Example: Show that the reflectance R// is equal to the amplitude reflection coefficient squared, r//2, and also show that R =r 2. By definition: Optics II----by Dr. H. Huang, Department of Applied Physics 19

The Hong Kong Polytechnic University Polarization Optical Activity: The property that causes the plane of polarization of P-state light to rotate when it passes through certain material. Viewing the beam head-on, if the rotation is clockwise, the material is referred to as dextrorotatory or d-rotatory. If the rotation is anticlockwise, the material is referred to as laevorotatory or l-rotatory. Phenomenologically, the optical activity can be explained by viewing that the linearly polarized light as to be the superposition of equal amounts of left- and right-circularly polarized components through an optically active materials with different velocities, v. L and v. R, respectively. Optics II----by Dr. H. Huang, Department of Applied Physics 20

The Hong Kong Polytechnic University Polarization Some Applications: Photoelasticity Kerr effect Homework: 12. 1; 12. 3; 12. 4; 12. 6; 12. 7; 12. 10 Optics II----by Dr. H. Huang, Department of Applied Physics 21