Advanced Test Equipment Rentals ATEC (2832)

Transcription

1 Established 98 Advanced Test Equipment Rentals ATEC (83) 9.5. THE AGILENT 869A POLARIZATION CONTROLLER A number of polarization controllers have been developed for the commercial market. Among the best known are the Agilent 869A and the Agilent 896A polarization controllers. The former provides deterministic coverage only for linearly polarized light whereas the latter completely covers the Observable Polarization Sphere non-determinstically; the meaning of these terms is discussed below. The Agilent 869A polarization controller consists of a built-in linear polarizer (LP) followed by a rotatable quarter-waveplate (QWP) and a rotatable halfwaveplate (HWP); each waveplate can be rotated independently of the other waveplate. The linear polarizer generates a linear state of polarization with a high extinction ratio. A schematic drawing of the LP-QWP-HWP controller is shown below. OPTICAL LINEAR QUARTER HALF SOURCE POLARIZER WAVEPLATE WAVEPLATE S Figure Optical configuration of the Agilent 869A polarization controller. In order to understand the behavior of this controller we determine the Stokes vector at the output of each polarizing element. We initially begin with the beam that emerges from the linear polarizer of arbitrary polarization and whose Stokes vector is cos α S = sin α 0 (9.5.) The Mueller matrix of the rotatable quarter-waveplate is M QWPROT cos θ cosθ sin θ sin θ = 0 cosθ sin θ sin θ cos θ 0 sinθ cosθ 0 (9.5.) where the subscript indicates that this is the rotation angle of the QWP. Multiplying (9.5.) by (9.5.) the Stokes vector of the beam emerging is S OUTQWP cos(α 4 θ ) + cos α = sin(α 4 θ ) + sin α sin(α θ ) (9.5.3)

2 We can immediately plot (9.5.3) on the Observable Polarization Sphere and we find S S S 3 S (a) (b) S (c) S 3 S (c) Figure Plot of the Stokes vector, (9.5.3), for input linearly polarized light propagating through a rotating quarter-waveplate. The views are (a) along the S S 3 axis, (b) along the axis, and (c) a non-axial view. The figures in Figure 9.5. clearly show the familiar figure 8s. We emphasize that the input beam must be linearly polarized (LP). Thus, Figure 9.5. shows that all polarization points (states) can be reached on the Observable Polarization Sphere using () any state of linearly polarized light and () a rotating quarterwaveplate. This behavior is confirmed by plotting each of the Stokes parameters in (9.5.3) as a 3D plot.

3 ( S ) S ( S3) ( ) Figure Plot of the Stokes polarization parameters S, S, and S, 3 respectively, in (9.5.3) showing that the extremes are m. The polarization sphere is completely covered. In order to obtain a specific polarization state the output polarized light described by (9.5.3) must be rotated to a specific state along the prime meridian. This can be accomplished by rotating a half-waveplate. The Mueller matrix for the rotated half-waveplate is M HWPROT cos4θ sin4θ 0 = 0 sin4θ cos4θ (9.5.4) where θ is the angle of rotation of the half-waveplate. The Stokes vector of the beam emerging from the waveplate pair is then obtained by multiplying (9.5.4) by (9.5.3) and we find

4 cos(α + 4θ 4 θ ) + cos(α 4 θ ) S ' = sin(α + 4θ 4 θ ) sin(α 4 θ ) sin(α θ ) (9.5.5) Equation (9.5.5) is now plotted on the Observable Polarization Sphere and we find: S S S 3 S S S 3 S Figure Plot of the Stokes vector, (9.5.5), of the beam emerging from the Agilent 869A polarization controller. The circles around the S 3 axis are indicative of the rotational behavior of the QWP-HWP configuration. The plots are made by incrementing the angles α, θ, and θ. Figure clearly shows that the Agilent 869A polarization controller covers the Observable Polarization Sphere for input linearly polarized light (LP). This is readily confirmed by also analyzing the Stokes vector, (9.5.5). It is not possible to plot (9.5.5) in 3D as we have been doing previously because there are three

5 variables, namely, α, θ, and θ. However, we can see, for example, that the coefficients of the two terms in the parameter S are / and / and the maximum and minimum values can add up to - and + for specific combinations of the angles. Similarly, the remaining two parameters also have the extreme values of m. We can confirm positively the coverage by again making 3D plots of the Stokes parameters in (9.5.5). In this case, however, because there are three variables we arbitrarily assume that the input linearly polarized light is L+45P and plot using only the two rotational angle variables. We then find that (9.5.5) yields (S) (S3) (S) Figure D plots of (9.5.5) which confirm the polarization coverage of the Observable Polarization Sphere. Equation (9.5.5) shows that by varying the angles of the linear polarizer and the two waveplates we can obtain any desired state of polarization. That is, we can deterministically specify the output polarization state. Conversely, given a desired output polarization state we can determine the required angles of the linear polarizer and rotation angles of the two waveplates. This direct and inverse behavior is said to be deterministic whereupon we say that the controller allows deterministic control.

6 APPENDIX SPHERICAL SPIRALS In Section 0.7 we saw that the approximate solution for the equations that describe the propagation of a polarized beam in a magnetized coil (a Faraday rotator) led to the appearance of spirals when plotted on the Observable Polarization Sphere. We now show that the Stokes vector for the approximate solution is a well-known description of the mathematical form for spirals plotted on a sphere known as loxodromes (Greek. loxos, oblique ; dromos, course ); they are also known as spherical spirals. A loxodrome is the path taken by a point which travels from the north pole to the south pole of a sphere while keeping a fixed angle with respect to the meridians. The curve has an infinite number of loops since the separation of consecutive revolutions gets smaller and smaller near the poles as we shall soon see. To obtain the equations for the loxodrome we first write the Stokes vector, (0.7.3), in normalized form, V B0 z + V B0 z S( z) = VB0z z cos V B0 z R + VB0z z sin + V B0 z R (A0.) For simplicity of notation we set t = z / R. Equation (A0.) then becomes V B0 R t + V B0 R t S( z) = VB0Rt cost + V B0 R t VB0Rt sin t + V B0 R t (A0.) Next, we set x = VB0Rtso (A0.) is now written as x + x St () = x cost + x x sin t + x (A0.3)

7 We now construct a right triangle shown in the following figure with sides in terms of the coefficients in terms of x in (A0.3) in order to express the coefficients trigonometrically. +x -x c Figure A0-. Right triangle to determine angular relations in (A0.3) From the triangle the following relations are constructed: x x sin c = + x (A0.4) x cosc = + x (A0.5) Substituting (A0.4) and (A0.5) into (A0.3) then yields sin c St () = cosccost coscsin t (A0.6) The terms within the Stokes vector, (A0.7), define the equations for a loxodrome: S = sin c (A0.7) S = cosccost (A0.8) S3 = coscsin t (A0.9) We now rewrite (A0.3) in a more convenient form by rewriting it as at + at St () = at cost + at at + at sin t (A0.0)

8 From (A0.4) and (A0.5) the parameter c can now be expressed by x at c = tan = tan x at (A0.) It is of interest to see the effect of the constant a on the behavior of c in (A0.); we recall that t = z/ R. Plots of (A0.) for various values of a are shown in the following figure (the upper curve corresponds to the smallest a value.) Figure A0-. Plots of (A0.) for a = 0. (uppermost curve), 0.5,.0, and.0. We now plot space curves for the Stokes vector, (A0.0). In order to emphasize the spiral behavior we choose a weak magnetic field, namely, B 0 = 0.0 T and a Verdet constant of.7 o /cm T and R = cm. We show two plots, one in the standard view and the other in a nonstandard view. LHP LHP LVP LVP Figure A0-3. Spherical spirals arising from the propagation through a Faraday rotator. The curves are initially at LHP and spiral down to LVP. However, LVP light is reached only after an infinite number of spirals has taken place.