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ps08soln - 1 Physics 316 Cornell University Solution for...

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1 Physics 316 Solution for homework 8 Spring 2005 Cornell University I. EXERCISE 1 As discussed in class, you send a vertically polarized beam through a tilted x 0 /y 0 analyzer, where the tilted y 0 axis makes an angle of θ to the vertical. If you send one beam through an ideal horizontal polarizer afterwords you get a probability P 1 for transmission. If you now place the horizontal polarizer behind the other beam that comes out of the x 0 /y 0 analyzer, the probability for transmission is P 2 . Show that P 1 = P 2 for all θ . P 1 θ P 2 θ The transmission amplitude in a x 0 /y 0 analyser for the output states | x 0 i and | y 0 i is h x 0 | y i = sin θ and h y 0 | y i = cos θ respectively. If a horizontal polarizer is placed at the output of the x 0 /y 0 analyser, the transmission probability is now P 1 = | h x | x 0 i h x 0 | y i | 2 and P 2 = | h x | y 0 i h y 0 | y i | 2 for the two possible outputs of the analyzer. Since h x | y 0 i = sin θ and h x | x 0 i = cos θ , we get finally that P 1 = P 2 = sin 2 θ cos 2 θ . II. EXERCISE 2 How can a right circular polarized light beam be changed to a left circular light be with as little loss of intensity as possible ? A right-circularly polarized light beam can be expressed as the sum of a linearly polarized light beam in x direction and a linearly polarized light beam in y direction, the later with a phase shift of - π/ 2. A left circularly polarized beam can be expressed the same way, but with a phase shift of π/ 2. To convert one into another, it suffices to give a phase shift of π to the y component. Thus, we only need to let the light beam go trough two λ/ 4-plates. III. EXERCISE 3 Given a photon beam in polarization state Ψ which is a pure but unknown state of liner or circular polarization. It is not known whether it is right or left handed nor is the direction of linear polarization known. Specify a method by which you can determine the polarization state. What is the maximum number of measurements that will be needed to determine the state ? We know that the state | Φ i is either equal to A | L i , A | R i or A | x i + B | y i , where A and B are real coefficients.
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