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Three Polarizer “Paradox”If the polarizers are opposed at a 90° angle, the polarized light from the first polarizer is stopped by the second. If a third polarizer is sandwiched between the two opposed polarizers at a 45° angle some light gets through the last polarizer.
A beam of unpolarized light illuminates a vertical polarizer and 50% of the light emerges vertically polarized. This light beam encounters a diagonal polarizer oriented at a 45 degree angle to the original vertical polarizer and 50% of it emerges as diagonally polarized light. Finally 50% of the diagonally polarized light passes a horizontally oriented polarizer. In other words 12.5% of the light illuminating the first vertical polarizer passes the final horizontal polarizer. However, if the diagonal polarizer sandwiched between the vertical and horizontal polarizers is removed, no light emerges form the final horizontal polarizer.Using the figure below vector algebra will be used to analyze this so-called "three-polarizer paradox." The paradox being that it is surprising that the insertion of the diagonal polarizer between crossed polarizers allows photons to pass the final horizontal polarizer.
If you take two crossed polarizers (for example, a horizontal and vertical one), no light will get through them. Yet when you insert a third polarizer between the two, oriented diagonally, then some photons make it through. How does adding that polarizer (which will block some photons) cause photons to get through?Say that the first polarizer is horizontal. Any photons that make it through that one are then horizontally polarized. If the vertical polarizer comes next, it will block all of these photons. When the diagonal polarizer is in place, however, it will let half of them through and these transmitted photons will then be diagonally polarized. When these diagonally polarized photons arrive at the vertical polarizer, now half of them will get through—they have no "memory" of ever having been horizontally polarized.
Dirac Three Polarizers ExperimentIn his 1930 textbook The Principles of Quantum Mechanics, Paul Dirac introduced the uniquely quantum concepts of superposition and indeterminacy using polarized photons.Dirac's examples suggest a very simple and inexpensive experiment to demonstrate the notions of quantum states, the projection or representation of a given state vector in another basis set of vectors, the preparation of quantum systems in states with known properties, and the measurement of various properties.QuoteAlbert Einstein said of Dirac and polarization,"Dirac, to whom, in my opinion, we owe the most perfect exposition, logically, of this [quantum] theory, rightly points out that it would probably be difficult, for example, to give a theoretical description of a photon such as would give enough information to enable one to decide whether it will pass a polarizer placed (obliquely) in its way or not." Maxwell's Influence on the Evolution of the Idea of Physical Reality...1931, Ideas and Opinions, p.270Any measuring apparatus is also a state preparation system. We know that after a measurement of a photon which has shown it to be in a state of vertical polarization, for example, a second measurement with the same (vertical polarization detecting) capability will show the photon to be in the same state with probability unity. Quantum mechanics is not always uncertain. There is also no uncertainty if we measure the vertically polarized photon with a horizontal polarization detector. There is zero probability of finding the vertically polarized photon in a horizontally polarized state.Since any measurement increases the amount of information, there must be a compensating increase in entropy absorbed by or radiated away from the measuring apparatus. This is the Ludwig-Landauer Principle.The natural basis set of vectors is usually one whose eigenvalues are the observables of our measurement system. In Dirac's bra and ket notation, the orthogonal basis vectors in our example are | v >, the photon in a vertically polarized state, and | h >, the photon in a horizontally polarized state. These two states are eigenstates of our measuring apparatus.The interesting case to consider is a third measuring apparatus that prepares a photon in a diagonally polarized state 45° between | v > and | h >.Dirac tells us this diagonally polarized photon can be represented as a superposition of vertical and horizontal states, with complex number coefficients that represent "probability amplitudes."Thus,| d > = ( 1/√2) | v > + ( 1/√2) | h > (1)Note that vector lengths are normalized to unity, and the sum of the squares of the probability amplitudes is also unity. This is the orthonormality condition needed to interpret the (squares of the) wave functions as probabilities, as first proposed by Max Born in 1927.When these complex number coefficients are squared (actually when they are multiplied by their complex conjugates to produce positive real numbers), the numbers represent the probabilities of finding the photon in one or the other state, should a measurement be made. Dirac's bra vector is the complex conjugate of the corresponding ket vector.It is the probability amplitudes that interfere in the two-slit experiment. To get the probabilities of finding a photon, we must square the probability amplitudes. Actually we must calculate the expectation value of some operator that represents an observable. The probability P of finding the photon in state |ψ> at location (in configuration space) r isP(r) = < ψ | r | ψ >No single experiment can convey all the wonder and non-intuitive character of quantum mechanics. But we believe Dirac's simple examples of polarized photons can teach us a lot. He thought that his simple examples provided a good introduction to the subject and we agree....We animated Dirac's idea of introducing an oblique polarizer between the two crossed polarizers A and B that are blocking all light. Adding this filter actually allows more photons to pass through, which is counter-intuitive.
Albert Einstein said of Dirac and polarization,"Dirac, to whom, in my opinion, we owe the most perfect exposition, logically, of this [quantum] theory, rightly points out that it would probably be difficult, for example, to give a theoretical description of a photon such as would give enough information to enable one to decide whether it will pass a polarizer placed (obliquely) in its way or not." Maxwell's Influence on the Evolution of the Idea of Physical Reality...1931, Ideas and Opinions, p.270
from section 4, Superposition and indeterminacy, pp, 12-14The general principle of superposition of quantum mechanics... requires us to assume that between these states there exist peculiar relationships such that whenever the system is definitely in one state we can consider it as being partly in each of two or more other states. The original state must be regarded as the result of a kind of superposition of the two or more new states, in a way that cannot be conceived on classical ideas. Any state may be considered as the result of a superposition of two or more other states, and indeed in an infinite number of ways. Conversely any two or more states may be superposed to give a new state. The procedure of expressing a state as the result of superposition of a number of other states is a mathematical procedure that is always permissible, independent of any reference to physical conditions, like the procedure of resolving a wave into Fourier components. Whether it is useful in any particular case, though, depends on the special physical conditions of the problem under consideration.QuoteEinstein and Schrödinger attacked the idea of superposition by arguing it would apply to macroscopic objects like Schrödinger's Cat being partly alive and partly dead[Our example deals] with states differing only with regard to the polarization.The nature of the relationships which the superposition principle requires to exist between the states of any system is of a kind that cannot be explained in terms of familiar physical concepts. One cannot in the classical sense picture a system being partly in each of two states and see the equivalence of this to the system being completely in some other state. There is an entirely new idea involved, to which one must get accustomed and in terms of which one must proceed to build up an exact mathematical theory, without having any detailed classical picture.When a state is formed by the superposition of two other states, it will have properties that are in some vague way intermediate between those of the two original states and that approach more or less closely to those of either of them according to the greater or less 'weight' attached to this state in the superposition process. The new state is completely defined by the two original states when their relative weights in the superposition process are known, together with a certain phase difference, the exact meaning of weights and phases being provided in the general case by the mathematical theory. In the case of the polarization of a photon their meaning is that provided by classical optics, so that, for example, When two perpendicularly plane polarized states are superposed with equal weights, the new state may be circularly polarized in either direction, or linearly polarized at an angle π/4, or else elliptically polarized, according to the phase difference.The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states, A and B, such that there exists an observation which, when made on the system in state A, is certain to lead to one particular result, a say, and when made on the system in state B is certain to lead to some different result, b say.What will be the result of the observation when made on the system in the superposed state? The answer is that the result will be sometimes a and sometimes b, according to a probability law depending on the relative weights of A and B in the superposition process. It will never be different from both a and b. The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states. The probability of a particular result for the state formed by superposition is not always intermediate between those for the original states in the general case when those for the original states are not zero or unity, so there are restrictions on the 'intermediateness' of a state formed by superposition.In this way we see that such a drastic departure from ordinary ideas as the assumption of superposition relationships between the states is possible only on account of the recognition of the importance of the disturbance accompanying an observation and of the consequent indeterminacy in the result of the observation. When an observation is made on any atomic system that is in a given state, in general the result will not be determinate, i.e., if the experiment is repeated several times under identical conditions several different results may be obtained. It is a law of nature, though, that if the experiment is repeated a large number of times, each particular result will be obtained in a definite fraction of the total number of times, so that there is a definite probability of its being obtained. This probability is what the theory sets out to calculate. Only in special cases when the probability for some result is unity is the result of the experiment determinate.The assumption of superposition relationships between the states leads to a mathematical theory in which the equations that define a state are linear in the unknowns. In consequence of this, people have tried to establish analogies with systems in classical mechanics, such as vibrating strings or membranes, which are governed by linear equations and for which, therefore, a superposition principle holds. Such analogies have led to the name 'Wave Mechanics' being sometimes given to quantum mechanics. It is important to remember, however, that the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory, as is shown by the fact that the quantum superposition principle demands indeterminacy in the results of observations in order to be capable of a sensible physical interpretation. The analogies are thus liable to be misleading. Fig.7. If a photon was initially prepared in the state | v > by polarizer A, it can also be represented as a superposition of states | d > and | d' >. There is thus a 50% chance of such a d photon getting through polarizer C. Photons passing through C can then be represented as in a superposition of states | v > and | h >. The | h > photons can easily pass through polarizer B, since it is turned to be horizontally polarized, blocking polarizer A's vertical polarized light...
Einstein and Schrödinger attacked the idea of superposition by arguing it would apply to macroscopic objects like Schrödinger's Cat being partly alive and partly dead
What is the best explanation for Three Polarizer “Paradox”?
A beam of unpolarized light illuminates a vertical polarizer and 50% of the light emerges vertically polarized. This light beam encounters a diagonal polarizer oriented at a 45 degree angle to the original vertical polarizer and 50% of it emerges as diagonally polarized light. Finally 50% of the diagonally polarized light passes a horizontally oriented polarizer. In other words 12.5% of the light illuminating the first vertical polarizer passes the final horizontal polarizer. However, if the diagonal polarizer sandwiched between the vertical and horizontal polarizers is removed, no light emerges form the final horizontal polarizer.Using the figure below vector algebra will be used to analyze this so-called "three-polarizer paradox." The paradox being that it is surprising that the insertion of the diagonal polarizer between crossed polarizers allows photons to pass the final horizontal polarizer.https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Quantum_Tutorials_(Rioux)/Quantum_Optics/268%3A_The_Three-Polarizer_Paradox
Alan Feldman, Ph.D. Physics, University of Maryland, College Park (1991)The middle polarizer rotates the plane of polarization. The result is that it is no longer perpendicular to the direction of the third polarizer, and thus some of it gets through.
Peter Hauge, Ph.D. Engineering & MathematicsLet’s demystify this.When linearly polarized light encounters a polarizing element (a polarizer) the amount transmitted is cos (a) where a is the angle between the light and the polarizer. Moreover the transmitted light becomes aligned with the polarizer.In particular, if a = 90 degrees, the light is extinguished. Two polarizers at right angles, therefore, extinguish light.Paradox: inserting a 45-degree intervening polarizer permits one-half of the light to pass:But a paradox it’s not. The intervening polarizer diminishes and rotates (by 45 degrees) the light that hits the final polarizer. The condition for extinguishing the light is thus removed.The effect of the three polarizers is the twice attenuation of the light by cos (45).Finally, cos(45) x cos(45) = 0.5. Thus one-half of the light is transmitted.
Alan CooperA polarizer doesn’t just remove all parts of the wave train that are not aligned with it. It does remove the parts that are perpendicular to it, but for parts at any other angle it converts them to its own direction by selecting the parallel component. I don’t think any of the discussions of vectors or classical electrodynamics or quantum theory should be particularly illuminating to anyone who is actually asking this question. So I will go with something much simpler to imagine (though maybe much harder to analyse in detail).Imagine shaking a vertical wave into a long rope or wire (eg a skipping rope or clothesline for example). Now imagine that the line passes through a slot between two slats. (I’d use fence boards as an example but they might be too thin unless we turned them sideways.) If the wave is aligned with the slot it will go through, but if the slot and the wave are perpendicular (and the slot has some significant depth along the direction of the wire) then the wave will be blocked. And if the wave starts out at some intermediate angle then some of it will get through. What gets through will be aligned with the slot (basically due to the wire being pushed sideways as a result of rubbing against the sides of the slot) but will have a reduced amplitude due to friction etc.Now, with the original slot blocking the wave, put another slot at 45 degrees between the source of the wave and the blocking slot. What comes out of the tilted slot will be a wave in the direction of this intermediate slot. When this new wave reaches the slot that originally blocked the wave completely, some of it will get through because the new wave is not perpendicular to the slot.If this doesn’t strike you as a paradox, then neither should the tilted polarizer effect.
Karl Brace, Computer Engineer, Computer-Aided Design Software at IntelThe “three polarizer paradox” only seems like a paradox if you think of polarizing materials like filters that block light of the wrong polarization. If polarizers did work that way, then passing through any set misaligned polarizers (even if they weren’t perpendicular) would indeed block all the light.But polarizers don’t work that way.When a light wave hits a material with free electrons, those electrons oscillate back and forth (pushed by the magnetic component of the electromagnetic wave), and the movement of the electrons, as with the oscillation of any charged particle, radiate an out of phase wave that cancels the incoming wave. This is why metals, which have lots of electrons that are free to move in any direction, are opaque even in very thin sheets.In the polarizing material, however, the electrons are constrained at the microscopic level to only move back and forth in one direction. So their motion only cancels out the component of the light in that alignment, allowing the component of the light that is “properly” aligned to pass through.But this doesn’t mean that misaligned light is canceled entirely. If polarized light hits the filter with, say, a 45 degree misalignment, it will still cause the free electrons to oscillate in the way that they are allowed. The combination of the original light and the radiation from the oscillating electrons creates a wave aligned with the polarizer!So although passing through a polarizer causes some loss of intensity if the light and polarizer were not initially aligned, unless the polarizer is perpendicular to the original polarization, some will come through.
Quote from: OPWhat is the best explanation for Three Polarizer “Paradox”?Define "Best"...A simple explanation is that passing light through a polarizer represents a measurement, and any measurement is likely to change the thing measured.
I also found simpler answers using classical physics in quora, essentially asserting that middle polarizer rotates the polarization of the light beam. My experiments using microwave as well as laser in visible spectrum agree with this.
If you consider polarisation as a classical electromagnetic phenomenon, there is no "paradox".
What makes me wonder is why it was called paradox in the first place if it is clearly explained by classical physics?
All of physics is either impossible or trivial. It is impossible until you understand it, and then it becomes trivial.
What makes me wonder is why it was called paradox in the first place if it is clearly explained by classical physics? Have previous generation physicists overlooked it?
From Dictionary.com“Paradox definition, a statement or proposition that seems self-contradictory or absurd but in reality expresses a possible truth.”
Explanations with wide applicability, hence can still explain some variations of the original experiment. For example, what the explanations say if the third polarizer has the same polarization as the first? Will the outgoing light be brighter, dimmer, or just the same as in original experiment?
Obviously the double slit experiment!
In classical explanation, light coming out of the second polarizer is polarized 45 degree, so the third polarizer (closest to receiver) will give the same intensity whether it's vertical or horizontal. Does QM predict the same result?
Quote from: hamdani yusuf on 09/10/2020 09:27:49In classical explanation, light coming out of the second polarizer is polarized 45 degree, so the third polarizer (closest to receiver) will give the same intensity whether it's vertical or horizontal. Does QM predict the same result? Sorry, missed this until another post triggered this.If you look at the wave function between the 45° and vertical polarisers inyou will see that if you reverse the order of the basis vectors in the kets then the probability remains unchanged if the final polariser is horizontal
The diagram seems to acknowledge that light passing through diagonal polarizer has diagonal polarization state. It makes subsequent reasonings produce the same result as classical theory which also acknowledges axis rotation by the diagonal polarizer. Classical theory which doesn't take axis rotation into account fails to give correct result.
As you say, any classical theory which assumes, as many people do, that a polarising filter just filters has to give the wrong answer.
I'm fairly sure that Maxwell's eqns (which are classical- neither QM, nor explicitly relativistic) will give the right answer.