Claims
- 1. A polarization controlling system for controlling the state of polarization of a light beam comprising:
a) a waveplate having:
a substantially constant retardance 2λ; a variable rotation axisŵ={circumflex over (x)} Cos(θ)Cos(φ)+ŷ Sin(θ)Cos(φ)+{circumflex over (z)} Sin(φ); andN variable retarders, N≧2, the retarders for being disposed sequentially along a propagation direction to receive the light beam, each variable retarder having a fast axis disposed at substantially 45° from the fast axis of each nearest variable retarder, each variable retarder having a variable retardance rx, x having an odd or even value between 1 and N, each retarder having an odd value x having a rotation quaternion X(rx)={Cos(rx/2), Sin(rx/2){circumflex over (x)}}, each retarder having an even value x having a component rotation quaternion Y(rx)={Cos(rx/2), Sin(rx/2)ŷ}; wherein a total rotation quaternionQ(X(r1)Y(r2)X(r3) . . . )= . . . X(r3)*Y(r2)*X(r1); andb) controlling means for controlling the retardances rx in dependence upon at least one solution to Q(X(r1)Y(r2)X(r3) . . . )= . . . X(r3)*Y(r2)*x(r1)={Cos(λ), Sin(λ)ŵ}.
- 2. A polarization controlling system for controlling the state of polarization of a light beam as defined in claim 1 wherein the controlling means includes voltage generating means for generating voltages in dependence upon the at least one solution of Q(X(r1)Y(r2)X(r3). . . )= . . . X(r3)*Y(r2)*X(r1)={Cos(λ), Sin(λ)ŵ}.
- 3. A polarization controlling system for controlling the state of polarization of a light beam as defined in claim 2 wherein the voltages are for applying to the variable retarders.
- 4. A polarization controlling system for controlling the state of polarization of a light beam as defined in claim 1 wherein:
a) N=2; and b) the at least one solution includesr1=2 sign(Sin(θ))arccos[Cos(λ){square root}{square root over (1+(tan(λ)Cos(θ)Cos(φ))2)}];r2=2 arctan(tan(λ)Cos(θ)Cos(φ)); and 16φ=-arcsin[-1+1+[tan(λ)Sin(2θ)]2tan(λ)Sin(2θ)];wherein sign(x)=−1 when x<0; sign(x)=0 when x=0; and sign(x)=1 when x>0.
- 5. A polarization controlling system for controlling the state of polarization of a light beam as defined in claim 2 wherein:
a) N=2; and b) the at least one solution includesr1=2 sign(Sin(θ))arccos[Cos(λ){square root}{square root over (1+(tan(λ)Cos(θ)Cos(φ))2)}];r2=2 arctan(tan(λ)Cos(θ)Cos(φ)); and 17φ=-arcsin[-1+1+[tan(λ)Sin(2θ)]2tan(λ)Sin(2θ)];wherein sign(x)=−1 when x<0; sign(x)=0 when x=0; and sign(x)=1 when x>0.
- 6. A polarization controlling system for controlling the state of polarization of a light beam as defined in claim 3 wherein:
a) N=2; and b) the at least one solution includesr1=2 sign(Sin(θ))arccos[Cos(λ){square root}{square root over (1+(tan(λ)Cos(θ)Cos(φ))2)}];r2=2 arctan(tan(λ)Cos(θ)Cos(φ)); and 18φ=-arcsin[-1+1+[tan(λ)Sin(2θ)]2tan(λ)Sin(2θ)];wherein sign(x)=−1 when x<0; sign(x)=0 when x=0; and sign(x)=1 when x>0.
- 7. A polarization controlling system as defined in claim 1 wherein the variable retarders include a liquid crystal cell.
- 8. A polarization controlling system as defined in claim 7 wherein the controlling means includes means for applying a voltage to the liquid crystal cell.
- 9. A polarization controlling system as defined in claim 1 wherein the variable retarders includes an electro-optical material.
- 10. A polarization controlling system as defined in claim 9 wherein the controlling means includes means for applying a voltage to the electro-optical material.
- 11. A polarization controlling system as defined in claim 1, including at least one fixed retardance retarder for modifying the state of polarization of the light beam, to compensate for an offset retardance in the variable retarders.
- 12. A polarization controlling system for controlling the state of polarization of a light beam comprising:
a waveplate having:
a substantially constant retardance 2μ; a variable rotation axis ŵ′={circumflex over (x)} Cos(θ)+ŷ Sin(θ); a first variable retarder having a first fast axis and a variable retardance u; a second variable retarder having a second fast axis oriented at an angle of substantially 45° from the first fast axis and a variable retardance v; a third variable retarder having a third fast axis oriented at an angle of substantially 0° from the first fast axis and a variable retardance w; and controlling means for controlling the retardances u, v and w wherebyu=w=arctan(Cos(θ)tan(μ)), and 19v=2 sign(Sin(θ))arccos{Cos (μ)Cos[arctan(Cos(θ)tan(µ))]}wherein sign(x)=−1 when x<0; sign(x)=0 when x=0; sign(x)=1 when x>0; and 0<2μ<π; wherein the first, second and third variable retarders are disposed for modifying the state of polarization of the light beam for controlling the state of polarization of the light beam.
- 13. A polarization controlling system for controlling the state of polarization of a light beam comprising:
a half waveplate having: a retardance of substantially 180°; a variable rotation axisŵ={circumflex over (x)} Cos(θ)+ŷ Sin(θ);a first variable retarder having a first fast axis and a variable retardance u1; a second variable retarder having a second fast axis oriented at an angle of substantially 45° from the first fast axis and a variable retardance u2; a third variable retarder having a third fast axis oriented at an angle of substantially 0° from the first fast axis and a variable retardance u3; and a fourth variable retarder having a fourth fast axis oriented at an angle of substantially 45° from the first fast axis and a variable retardance u4; and controlling means for controlling the retardances u1, u2, u3 and u4 wherebyu1=2sign(Sin(θ))arccos [Cos(λ){square root}{square root over (1+(tan(λ)Cos(θ)Cos(φ))2)}];u2=2 arctan(tan(λ)Cos(θ)Cos(φ))+arctan[Cos(θ+Δθ)tan(μ)]; 20u3=2 sign(Sin(θ+Δ θ))×arccos{Cos (μ)Cos[arctan[Cos(θ+Δ θ)tan(μ)]]};u4=arctan[Cos(θ+Δθ)tan(μ)];wherein: 21φ=-arcsin[-1+1+[tan(λ)Sin(2θ)]2tan(λ)Sin(2θ)];Δθ=−arctan(Sin(φ)tan(λ)); 22µ=arctan[1Sin2(λ)Cos2(φ)-1];and0<λ<π/2; wherein the first, second, third and fourth variable retarders are disposed for modifying the state of polarization of the light beam for controlling the state of polarization of the light beam.
- 14. A polarization controlling system as defined in claim 12 wherein μ is substantially equal to π/4 radians.
- 15. A polarization controller for controlling the state of polarization of a light beam comprising first and second polarization controlling system as defined in claim 14.
- 16. A polarization controller as defined in claim 15, additionally comprising a polarization controlling system as defined in claim 13.
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] This application claim priority of U.S. Provisional Patent Application No. 60/425,529 filed Nov. 12, 2002, entitled “Endless Polarization Control,Algorithm Using Adjustable Linear Retarders With Fixed Axes” which is incorporated herein by reference for all purposes.
Provisional Applications (1)
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Number |
Date |
Country |
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60425529 |
Nov 2002 |
US |