EP421 Assignment 4: Polarization II: Applications of Optical Anisotropy use of the Jones Calculus (Handed Out: Friday 1 November 2013 Due Back: Friday 8 November 2013) 1. Optic Axis of Birefringent Crystals [D] indicates a Design Question Many crystalline materials which are useful in optics are uniaxial - that is the index ellipsoid (or indicatrix ) is elongated (or compressed) along one axis (typically taken to be the z axis) and has a circular cross- section about that axis. As we saw in class and in the textbook (Lipson 3 Chapter 6), such materials exhibit birefringence, allowing propagation of both an ordinary (o) as well as an extraordinary (e) ray. The following table lists the ordinary and extraordinary refractive indices for a few useful birefringent optical crystals Material n o n e Lithium niobate (LiNbO 3 ) 2.30 2.21 Calcite (CaCO 3 ) 1.66 1.49 KDP (potassium dihydrogen phosphate KH 2 PO 4 ) 1.51 1.47 Lithium niobate and KDP are used in electro- optic modulators (EOMS) for laser beams and optical communications (values of KDP refractive index above were taken from the website of Cleveland Crystals Inc., Highland Heights, OH, USA). For each of these materials: Write the dielectric tensor in principal axis form, e.g.:! ε 1 0 0 # ε = # 0 ε 1 0 # "# 0 0 ε 3 (Note you may assume the entries in the tensor ε are normalized by the permittivity of free- space ε0, i.e. D = ε 0 ε E, so that the entries in the dielectric tensor ε are dimensionless pure numbers) Sketch the index ellipsoid (a qualitative hand sketch is fine, but try to get the relative axes approximately scaled right). Align the extraordinary index with the z- axis for consistency. Sketch the elliptical cross- section of the ellipsoid created by the intersection of the ellipsoid with the x- z plane. Again a hand- sketch is fine, but try to get the scales approximately right.
Note that all 3 crystals above have ne < no they are therefore referred to as negative uniaxial. A positive uniaxial crystal would have ne > no. One example of this is zircon (zirconium silicate ZrSiO4), which has no = 1.923 and ne= 1.968. Write the dielectric tensor for zircon and sketch the index ellipsoid and elliptical cross section, as done for the materials above. Qualitatively, what is the difference between the index ellipsoids for negative uniaxial and positive uniaxial crystals? 2. [D] Quarter- Wave Plate Design: (a) Derive an expression for the minimum thickness of a Quarter- Wave Plate (QWP) made from materials with a refractive index difference δn = n S n F between the indices for the slow and fast waves. Show your steps. (b) You are asked to design a QWP for operation in the middle of the visible range (you can ignore any dispersion effects for now). This will involve specifying both a material and a thickness. The manufacturer can grind ceramic dielectrics down to a thickness of not less than 0.1 mm, with a precision of 0.002 mm (=2 μm). Design a QWP which this manufacturer can build. You need to specify (i) the plate material (ii) the optic axes chosen (iii) the plate thickness. Use the materials properties given in Table 1 below. (c) What is the phase shift error on this plate, given the manufacturing tolerance stated in part (c) Table 1. Refractive Indices of common biaxial and uniaxial crystals at λ=589 nm) (Data from Lipson, Lipson Lipson, Optical Physics, 4 th ed., Table 6.1 p. 180) n 1 ( n 0 ) n 2 n 3 (n e ) Mica 1.582 1.587 1.552 Lithium borate 1.578 1.601 1.621 Lithium niobate 2.30 2.21 Calcite 1.66 1.49 Sapphire (Al 2 O 3 ) 1.768 1.760 3. [D] Polarizing Prism Design: A Glan- type polarizer is a device made from two calcite prisms separated by an air gap. Calcite (a crystalline form of CaCO3, also called Iceland Spar) is a uniaxial birefringent crystal with refractive indices:
no = 1.66 (for the ordinary ray) ne = 1.49 (for the extraordinary ray) We can use this information to design several types of polarizing prism. Proceed as follows: (a) Calculate the critical angle for total internal reflection for the ordinary (o) ray emerging from the prism (b) Calculate the critical angle for total internal reflection for the extraordinary (e) ray emerging from the prism (c) Based on the above calculated angles, and with reference to Fig. 1 below, at what angle should the hypotenuse of the prism be cut (relative to the horizontal base) to separate the two polarization components of the beam using total internal reflection? Figure 1: Schematic of Glan-Foucault prism (also called a Glan-Air prism), showing the air gap (figure by Bob Mellish. Wikimedia commons. Reproduced under GNU Free Doumentation License/Creative Commons Share- Alike 3.0 Unported) (d) The air- gap polarizing prism you designed above is called a Glan- Foucault or Glan- Air polarizing prism (Fig. 1) and is useful for separating polarization components of powerful laser beams. Fig. 2 shows a variant of this same idea in which two calcite prism halves are cemented together with an index- matching transparent optical cement which has its refractive index chosen to be n= 1.49 in order to match ne for calcite. In this case there will be no refraction of the extraordinary ray. Using the same idea as above, choose a new hypotenuse angle (relative to the horizontal base) to separate the o ray by total internal reflection (this design is called a Glan- Thomson polarizing prism)
Figure 2: Glan-Thompson prism (cemented): figure by Bob Mellish. Wikimedia commons. Reproduced under GNU Free Doumentation License/Creative Commons Share-Alike 3.0 Unported) 4. Use of Jones Matrix Methods: Solve the following problems using the Jones calculus method (See Appendix for the necessary Jones vectors and Jones matrices) (a) Polarization rotator: Using appropriate trigonometric identities, find the Jones matrix for a polarization rotator which will rotate the plane of polarization of an incident linearly polarized beam by angle β. (b) Let the slow axis of a quarter- wave plate (QWP) correspond to the Ex- axis (therefore the fast axis corresponds to the Ey axis). Now consider a polarizer P placed before the QWP. Let the transmission axis of this polarizer make angle θ with respect to the x- axis, and find the Jones vector which describes the polarization state of the light after emerging from the QWP. It should be elliptically polarized. What is the ratio of the Ey/Ex electric field amplitudes, as a function of θ? At what angle θ (wrt. the x- axis) do you have to set the transmission axis of P to obtain Ey/Ex = ½? (c) Suppose the electric field of a linearly polarized incident light beam of amplitude E0 makes angle α with the x- axis. Write the Jones vector for this polarized beam. Now imagine a half- wave plate (HWP) is inserted in the beam path. Use Jones calculus to find the result polarization state if the slow axis of the HWP is aligned with the x- axis. What if the effect on the polarization state (in words)? Now repeat for the case when the fast- axis of the HWP is aligned along the x- axis. Can you think of an application for this? 5. Optical Isolator: Consider an optical isolator consisting of a linear polarizer P followed by a quarter- wave plate QWP with slow axis oriented at 45 to the transmission axis of P. What is the Jones vector of the light after the QWP? If the light undergoes reflection and is transmitted back through the device, what happens? (Use the Jones calculus method to analyze this). 6. Optical Activity of Corn Syrup (Lipson 3 problem 6.8): Corn syrup is an optically active medium. In a simple corn syrup polarimetry experiment, white light first passes through a polarizer P and then a transparent sample holder filled with corn
syrup. The path length of the light in the corn syrup is 10 cm. At the far end is an analyzer polaroid A. As the analyzer is rotated through 90 the colour of the light emerging from the analyzer goes through the full visual spectrum from red ~ 650 nm to blue ~ 400 nm. Use this information to estimate the refractive index difference between left- handed (L) and right- handed (R) polarization states (nl nr) for corn syrup (assume the mean refractive index of corn syrup is n ~ 1.5) Appendix: Jones Calculus Info: Jones Vectors! 1! 0! Horizontal (x) Polarization: # Vertical (y) Polarization: # General: # " 0 " 1 " 1! 1 Left- Hand Circ. Pol. (LHCP): # Right- Hand Circ. Pol. (RHCP): 2 " i Jones Matrices! 1 0! 0 0 1! Horizontal # Vertical # 45 # " 0 0 " 0 1 2 " 1 1 1 1 1 " 2 # Quarter- Wave Plate QWP (top sign- - slow axis vertical, bottom sign- - slow axis horizontal) e iπ! 4 1 0 # " 0 ±i 1 i ' cosα sinα Half- Wave Plate (HWP) (top sign- - slow axis vertical, bottom sign- - slow axis horizontal) e iπ " 2 1 0 ' # 0 1