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Chapter 8: Problem 34

Polarization "projection operator." If a piece of linear polaroid with easy axis along \(\hat{x}\) is placed in a beam of light containing a mixture of all sorts of polarization, the polaroid absorbs all light that does not have linear polarization along \hat{x} . ~ I t ~ has an "output" at the rear of the polarizer consisting of light linearly polarized along \hat?. We shall call this piece of polaroid a "projection operator." It "projects out" the x polarization without loss (neglecting small reflections) and delivers it at its output end. Note that this " \(x\) projection operator" can be run either forward or backward; i.e., either face of the polaroid may be used as the input end. Now consider a piece of circular polarizer consisting of a piece of linear polarizer (input end) glued to a quarter-wave plate with optic axis at 45 deg to the easy axis of the polaroid. This polarizer puts out (for example) right-handed light. But it absorbs half of any righthanded light incident. If it is run backward, it passes incident right-handed light and absorbs left-handed light. But when it thus passes right-handed light incident on the quarter-wave-plate face, it delivers it out the polaroid face as linearly polarized light. Therefore it is not what we are calling a polarization projection operator. Here is the problem: Invent circular polarization projection operators, one for left- handed and one for right-handed light. The right-handed projection operator should transmit incident right-handed light with no loss (neglecting small reflections) and should deliver it as right-handed light. It should absorb left-handed light. Question: Is your circular polarization projection operator reversible? Can you use either face for the input end?

Short Answer

Expert verified
The circular polarization projection operator is not reversible.

Step by step solution

01

Understanding the Problem

A polarization projection operator selectively transmits certain polarizations and absorbs others. The task requires designing a circular polarization projection operator that transmits either left-handed or right-handed polarized light fully, while absorbing the opposite polarization, and determining if such a system is reversible.
02

Analyzing Existing Circular Polarizers

A standard circular polarizer consists of a linear polarizer followed by a quarter-wave plate and absorbs half of the incident circularly polarized light when used conventionally. When reversed, it performs selectively based on the handedness of incident light but modifies the output polarization to linear as opposed to circular.
03

Designing the Right-Handed Polarization Projection Operator

To create a right-handed projection operator, we need a setup where right-handed circularly polarized light passes through without loss, and left-handed light is absorbed. This can be achieved by using a quarter-wave plate first to convert right-handed light to linear polarization along the polaroid's axis, allowing it to pass, and converting left-handed light to orthogonal linear polarization, resulting in absorption by the polaroid.
04

Designing the Left-Handed Polarization Projection Operator

Similarly, for a left-handed projection operator, a quarter-wave plate placed first converts left-handed light to linear along the polaroid's axis, which passes through the polaroid, while right-handed light is blocked due to resulting linear polarization orthogonal to the polaroid's axis.
05

Reversibility of the System

The system is not reversible. Each operator functions only when the quarter-wave plate is correctly positioned relative to the polaroid for the specific handedness of input light. When run backward, it does not maintain the desired selective polarization projection without altering polarization.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Polarization
Linear polarization refers to the orientation of oscillations in a light wave along a single direction. This is much like how a skipping rope moves side-to-side when shaken. For instance, a linear polarizer allows light to pass only if its electric field oscillates in the direction aligned with the polarizer's axis.

  • Light, when passed through a linear polarizer, emerges with its electric component oscillating along the polarizer's "easy axis," filtering out all other components.
  • A linear polarizer can block 50% or more of randomly polarized light, depending on their initial polarization axes.
  • This linear polarization process is crucial in applications like sunglasses and camera filters, where reducing glare is needed.
Circular Polarization
Circular polarization occurs when the electric field of light describes a helix along the direction of travel, rotating in either a right-handed or left-handed manner. You can think of it as if the end of the vector tracing the electric field describes a circular motion.

  • To impart circular polarization, an optical device called a quarter-wave plate is used, placed diagonally at 45掳 relative to the light's original linear polarization.
  • Right-handed circularly polarized light rotates clockwise, while left-handed rotates counter-clockwise as perceived from the source.
  • This polarization type is useful in areas such as 3D cinema and optical communication, offering unique data transmission channels.
Polarization Projection Operator
A polarization projection operator enables the passage of light polarized in a specific orientation while absorbing the rest. Analogous to a sieve, it allows only certain parts through. This is used to manage light polarization efficiently.

  • In linear polarization projection, only light waves aligned with the polarizer's orientation pass through completely.
  • Designing a circular polarization projection operator is challenging since it must filter based on the circular orientation without converting it to linear polarization.
  • These operators require precise alignment to ensure selective transmission based on the light's polarization state.
Optics
Optics is the branch of physics that deals with the behavior and properties of light, including its interactions with matter. It covers everything from lenses and mirrors to prisms and optical fibers. Understanding optics is fundamental to designing devices like polarization projection operators.

  • Optics explores phenomena such as refraction, reflection, dispersion, and polarization.
  • The principles of optics are essential in developing numerous technologies, including eyeglasses, cameras, and optical instruments.
  • Polarization, a key topic within optics, plays crucial roles in applications that require manipulation of light properties, like enhancing contrast or selectively transmitting light.

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Most popular questions from this chapter

Circularly polarized light of intensity \(I_{0}\) is incident on a sandwich of three polaroids. The first and third polaroids are crossed, i.e., their easy axes are at 90 deg to one another. The middle polaroid makes an angle \(\theta\) with the axis of the first polaroid. Show that the output intensity is \(\frac{1}{2} I_{0} \cos ^{2} \theta \sin ^{2} \theta\)

Phase relations in specular reflection of light from glass. We are trying to verify the relationships shown in Fig. 8.8. The experimental setup is as in Home Exp. 8,25, except that between the piece of glass lying on the table and the light we place a piece of polaroid with easy axis rotated at 45 deg to the horizontal. (Sugges tion: For a convenient mounting, put a little nonhardening putty or glazing compound on a microscope slide and stick the corner of the polaroid into the putty. The microscope slide can be the glass surface used for reflecting the light.) Suppose that when your eye is at the microscope slide and you are looking back at the light bulb through the first polaroid, the easy direction of the polaroid is "upper right to lower left." Keep analyzing the polarization of the reflected light as you vary the angle of incidence by changing the position of the slide or light bulb. You will find that near normal incidence the polarization of the reflected light is along "upper left to lower right." As you move the slide and approach Brewster's angle, the polarization remains linear, but it rotates toward the horizontal. It becomes horizontal at Brewster's angle and then continues to rotate in the same direction as you go beyond Brewster's angle toward grazing incidence; i.e., it becomes "lower left to upper right." Thus in going from normal to grazing incidence, the polarization rotates by \(90 \mathrm{deg}\), as predicted by Fig. 8.8. (At normal incidence, both components are equally well reflected, as they must be since they don't know which is which, so to speak. Therefore the polarization is at 45 deg. At grazing incidence, both components are almost completely reflected, so that they are equally well reflected. Therefore the polarization is at 45 deg.) It is interesting to notice that the polarization remains linear through all of this experiment. This means that there are no phase shifts other than 0 or 180 deg introduced between those components in the plane of incidence and those perpendicular to it. Thus the incident waves always experience a purely resistive load when they reflect. That is what we expect for reflection from a transparent medium.

Suppose that a beam of linearly polarized light is incident on a half-wave plate which is rotating about the beam axis with angular velocity \(\omega_{0}\). Show that the output light is linearly polarized, with the polarization direction rotating at \(2 \omega_{0}\) -

Optical activity. Suppose you send linearly polarized light through a length \(L\) of Karo corn syrup and find that, for \(L=5 \mathrm{~cm}\), red light is rotated by 45 deg. Now reflect the light that has passed through the syrup from a mirror and send it back through the syrup, so that the total length is \(10 \mathrm{~cm}\). (If you do the experiment, make the angle of reflection not quite \(180 \mathrm{deg}\); then look at the "image light bulb" through both the "real syrup" and the "image syrup." As a control experiment, you can look through the "image syrup" alone by moving your head.) Question: After the two traversals, is the linear polarization at 0 or 90 deg to the original direction?

Slinky polarization. Find a slinky and a partner. You and your partner hold opposite ends of the slinky. (a) Let each shake the slinky in a clockwise circular rotation (from his own point of view). If this doesn't convince you that linear polarization is a superposition of opposite circular polarizations, nothing else will. (b) With each person using a book as a straightedge to guide his hand, let one partner shake linear polarization at \(45 \mathrm{deg}\) to the horizontal, and let the other shake linear polarization at 90 deg to the first. (The 45 deg angle is so as to prevent gravity from giving a big asymmetry.) One counts out loud, " \(1,2,3,4,1,2,3,4, \ldots "\) four beats to a cycle, or perhaps four to a half-cycle), with "one" coming at a reproducible phase of the motion of his hand. The other shakes in phase, or 180 deg out of phase, or 90 deg out of phase. It takes some concentration not to be distracted by what you see. ( \(c\) ) With the far end fixed to something (your partner can now go home), shake out a circularly polarized wave packet of one or two t?ns. Verify that it conserves angular momentum upon reflection. Verify that if the angular momentum is along the propagation direction, the shape is that of a left- handed screw, and that the handedness reverses upon reflection.
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