Reversing the angular momentum of a photon is not a problem.  In fact, a half-wave plate does reverse the helicity of circularly polarized light. If plane polarized light passes through a sugar solution, reflects from a mirror, and then passes back through the same solution, the final plane of polarization is the same as the initial one.  If the plane rotates clockwise in its initial passage, it must do the same in the final passage, but it's going in the opposite direction.  A sugar solution is isotropic; it doesn't care about the propagation direction of the light. On the other hand, in a Faraday rotator, if the plane rotates CW when propagating parallel to the magnetic field, it rotates CCW when propagating antiparallel to the field.  The emerging light will have a different polarization plane.  This is why optical isolators (devices which transmit light in one direction, block it in the other direction) use Faraday rotators. But using a Faraday rotator to rotate the plane twice by 45 degrees isn't good enough to solve the puzzle.  We have a polarization meter.  When measuring the incident light, its x = horizontal axis is east and its y = vertical axis is up.  It tells us that the incident light is polarized at angle theta.  To measure the reflected light, we swing the meter around, rotating it by 180 degrees about some axis, say the vertical.  Now the x axis points west and the y axis is still up.  The initial polarization plane is now -theta.  The emerging polarization plane, having been rotated 90 degrees, is theta' = 90-theta.  But if theta = 45, we have theta' = theta, and the polarization has not been changed to the opposite state, as the puzzle requires.   --  Gene From: Warren D Smith <warren.wds@gmail.com> To: math-fun@mailman.xmission.com Sent: Monday, June 1, 2015 5:16 PM Subject: [math-fun] Salamin polarization puzzle Here is a puzzle concerning the optics of polarized light. Every state of polarization has its opposite.  For linear polarization, it's linear but rotated 90 degrees.  For ere is a puzzle concerning the optics of polarized light. Every state of polarization has its opposite.  For linear polarization, it's linear but rotated 90 degrees.  For circular polarization, it's circular with opposite helicity.  For general elliptic polarization, it's elliptic with the ellipse rotated 90 degrees, and the helicity reversed.  On the Poincaré sphere, opposite states of polarization are represented by diametrically opposite points. The puzzle is to construct an optical device that reverses the polarization state.  For any input, the output is the opposite polarization.  Or, prove that it can't be done.   --  Gene.  For general elliptic polarization, it's elliptic with the ellipse rotated 90 degrees, and the helicity reversed.  On the Poincaré sphere, opposite states of polarization are represented by diametrically opposite points. The puzzle is to construct an optical device that reverses the polarization state.  For any input, the output is the opposite polarization.  Or, prove that it can't be done.   --  Gene --WDS: There is a sense in which it cannot be done for a flow-thru device. For linear polarization, it can be done, you just rotate the plane of polarization by 90 degrees by passing it thru a sugar solution; also faraday effect works.   But when you said "for circular polarization, it's circular with opposite helicity" that seems to be the death blow.  Because the input is photons with clockwise spin while the output is photons with anticlockwise spin.  This means, in view of conservation of angular momentum, your device will be forced into rotation, faster the longer you run it.   I.e. it is applying torque to your light, and the light is applying torque to it. So this is not, and cannot be, a "passive device" of the usual 100% efficiency optical variety. Having said that, maybe this does not bother you. And anyhow here is a "back at you" device: An ordinary mirror in combination with sugar solution layer of the correct thickness will input a light beam (normal to mirror)  and output one (reflected) that is both circularly helicity-reversed  and also linear polarization rotated 90. Does that satisfy you?