Consider, now, putting such a wave plate between two crossed polarizers, oriented at \(\pm 45^{\circ}\), as shown in Figure \( 12.5\). Without the wave plate, no light would get through because the first polarizer transmits only light polarized at \(45^{\circ}\), described by the \(Z\) vector \[Z=\left(\begin{array}{l} 1 / \sqrt{2} \\ 1 / \sqrt{2} \end{array}\right)\]

A “polarizer” is a device that allows light polarized in a particular direction (the “easy transmission axis” of the polarizer) to pass through with very little absorption, but absorbs most of the light polarized in the perpendicular direction. Thus an unpolarized light beam, passing though the polarizer, emerges polarized along the easy axis.

Figure \( 12.5\): Initially unpolarized light passing through a pair of crossed polarizers with a wave plate in between.

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We can now use our matrix language to see how this leads to optical activity. Up to an irrelevant overall phase, we can choose the phase produced on the left-circularly polarized light to be −\(\theta\) and that on the right-circularly polarized light to be \(\theta\). Then we can represent the action of the syrup on an arbitrary wave by the matrix \[e^{-i \theta} P_{+}+e^{i \theta} P_{-} ,\]

Clearly, the optical activity of corn syrup cannot depend on crystal structure, because the stuff is a perfectly uniform liquid, completely invariant under rotations in three-dimensional space. It can have no special axes, or any such thing. Optical activity must work very differently from birefringence.

For electromagnetic waves, the most familiar example of a polarizer, Polaroid, was invented by Edwin Land over 50 years ago, partly in experiments done in the attic of the Jefferson Physical Laboratory, where he worked as an undergraduate at Harvard. The idea of polaroid is to make material that conducts electricity (poorly) in one direction, but not in the other. Then the electric field in the conducting direction will be absorbed (the energy going to resistive loss), while the electric field in the nonconductive direction will be unaffected. One way of doing this is to make sheets of polymer (polyvinyl alcohol) stretched (to align the polymer molecules along a preferred axis) and doped with iodine (to allow conduction along the polymer molecules).2

For frequencies such that \(e^{-i \Delta \phi}\) is −1, the light is polarized in the −\(45^{\circ}\) direction, and gets \(e^{-i \Delta \phi}\) through the second polarizer without further attenuation. But for frequencies such that e is 1, the light is still absorbed by the second polarizer. Intermediate frequencies are partially absorbed.

Note that in general the phase difference, \(\Delta \phi\), depends on the frequency of the light. Even if \(n_{x}\) and \(n_{y}\) depend on frequency, it would be a bizarre accident if that dependence canceled the \(\omega\) dependence from the explicit factor of \(\omega\) in (12.39).

One reason that polarization is important is that the polarization state of an electromagnetic wave can be easily manipulated. Two of the most important devices for such manipulation are polarizers and wave plates.

“Wave plates” are optical elements that change the relative phase of the two components of \(Z\). Wave plates are possible because there are materials in which the index of refraction depends on the polarization. This property is called “birefringence.” It can happen in various ways.

where \(P_{\pm}\) are matrices that pick out the left- and right-circularly polarized components, respectively. They satisfy \[P_{\pm}\left(\begin{array}{c} 1 \\ \pm i \end{array}\right)=\left(\begin{array}{c} 1 \\ \pm i \end{array}\right), \quad P_{\pm}\left(\begin{array}{c} 1 \\ \mp i \end{array}\right)=0 .\]

A wave plate in which the phase difference is \(\pi / 2\) is called a “quarter wave plate.” For a wave plate in which the phase difference is between 0 and \(\pi\), it is conventional to call the axis with the smaller phase the “fast axis.” A quarter wave plate with fast axis along the 1 axis is represented by \[Q_{0}=\left(\begin{array}{ll} 1 & 0 \\ 0 & i \end{array}\right) .\]

Here are two amusing devices that you can make with these optical elements (or matrices). Consider the combination of first a polarizer at \(45^{\circ}\) and then a quarter wave plate, as shown in Figure \( 12.6\). By forming the matrix product, \(Q_{0} P_{\pi / 4}\), you can see that this produces counterclockwise circularly polarized light from anything with a component of polarization in the \(\pi / 4\) direction. The argument goes like this. The product is \[Q_{0} P_{\pi / 4}=\left(\begin{array}{cc} 1 & 0 \\ 0 & i \end{array}\right)\left(\begin{array}{cc} 1 / 2 & 1 / 2 \\ 1 / 2 & 1 / 2 \end{array}\right)=\left(\begin{array}{cc} 1 / 2 & 1 / 2 \\ i / 2 & i / 2 \end{array}\right) .\]

If wavelengths of light in the range of 11–14 nm are used, it is possible to construct reflecting optics of moderate efficiency (> 60%) using multilayer films. This opens up the possibility of projection optics and reduction imaging. With a numerical aperture of 0.25, a wavelength of ~13.5 nm (this particular choice is explained shortly), and a k1 value of 0.6, the resolution is

However the birefringence is produced, we can make a wave plate by orienting the material so that the \(x\) and \(y\) directions correspond to different indices of refraction, \(n_{x}\) and \(n_{y}\), and then making a slice of the material in the form of a plate in the \(x\)-\(y\) plane, with some thickness \(\ell\) in the \(z\) direction. Now an electromagnetic wave traveling in the \(z\) direction through the plate has different \(k\) values depending on its polarization: \[k=\left\{\begin{array}{l} \frac{n_{x}}{c} \omega \text { for polarization in the } x \text { direction } \\ \frac{n_{y}}{c} \omega \text { for polarization in the } y \text { direction } \end{array}\right.\]

For the transverse oscillations of a string, a polarizer is simply a slit that allows the string to oscillate in one transverse direction but not in the perpendicular direction.

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The object \(P_{\theta}\) is called a “projection operator,” because it projects the vector onto the direction parallel to \(u_{\theta}\). It satisfies \[P_{\theta} P_{\theta}=P_{\theta},\]

This is just the rotation matrix \(R_{\theta}\), of (12.34)! \(R_{\theta}\) rotates both components of any light by an angle \(\theta\).

every plane wave is polarized. However, in an “unpolarized” beam, the light wave consists of a range of angular frequencies with different polarizations. As a result of the interference of the different harmonic components of the wave, the polarization wanders more or less randomly as a function of time and space, and on the average, no particular polarization is picked out. A simple example of what this looks like is animated in program 12-2, where we plot an electric field of the form \[\begin{aligned} &E_{x}(t)=\cos \left(\omega_{1} t+\phi_{1}\right)+\cos \left(\omega_{2} t+\phi_{2}\right), \\ &E_{y}(t)=\cos \left(\omega_{3} t+\phi_{3}\right)+\cos \left(\omega_{4} t+\phi_{4}\right), \end{aligned}\]

In particular, the phase difference, between \(x\) and \(y\) polarized light in going through the plate is \[\Delta \phi=\frac{n_{x}-n_{y}}{c} \omega \ell .\]

This should convince you that in general if the fast axis is in the \(\theta\) direction, the quarter wave plate looks like \[Q_{\theta}=P_{\theta}+i P_{\theta+\pi / 2} .\]

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In any beam of light, at any given point and time, the electric field points in a particular direction. Likewise, because any plane electromagnetic wave with a definite angular frequency can be described by (12.20) and (12.21), \[\begin{aligned} \vec{E} &=\operatorname{Re}\left(\vec{e}(\vec{k}) e^{i \vec{k} \cdot \vec{r}-i \omega t}\right) \\ \vec{B} &=\operatorname{Re}\left(\vec{b}(\vec{k}) e^{i \vec{k} \cdot \vec{r}-i \omega t}\right) \end{aligned}\] \[\vec{b}(\vec{k})=\frac{1}{\omega} \vec{k} \times \vec{e}(\vec{k})=\frac{n}{c} \hat{k} \times \vec{e}(\vec{k}) \quad \text { and } \quad \hat{k} \cdot \vec{e}(\vec{k})=0.\]

But compare this with the apparently similar situation in which the second polarizer transmits light polarized at \(45^{\circ}\) in the \(x\)-\(y\) plane, as shown in Figure \( 12.9\). Now the wave description tells us that the intensity in region \(III\) is reduced by another factor of 2 from that in region \(II\). This is impossible to interpret in terms of classical particles. To see this, it is only necessary to turn down the intensity so that only one photon comes through at a time. Then the first polarizer is OK. As before, if the photon is polarized in the \(x\) direction, it get through. But now what happens at the second polarizer. The photon cannot split up. Either it gets through or it doesn’t. To be consistent with the wave description, in which the intensity is reduced by another factor of two, the transmission at the second polarizer must be a probabilistic event. Half the time the photon gets through. Half the time it is absorbed. There is no way for the

One might wonder about the reason for the handedness of the sugar molecules. In fact, there are physical processes, the weak interactions that give rise to \(\beta\)-radioactivity, that look different when reflected in a mirror3 and thus in principle could distinguish between left-handed and right-handed molecules. However, these interactions are most likely irrelevant to the handedness of corn syrup. Probably, the reason is biology rather than physics. Long ago, when the beginnings of life emerged from the primordial soup, purely by accident, the right-handed sugars were used. From then on, the handedness was maintained by the processes of reproduction.

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The same effect may arise because of the inherent structure of a transparent crystal. An example is the naturally occurring mineral, calcite, a crystalline form of calcium carbonate, \(CaCO_{3}\). Crystals of calcite have the fascinating property of splitting a beam of unpolarized light into its two polarization states. Birefringence can even be produced mechanically, by stressing a transparent material, squeezing the electronic structure in one direction.

The discussion of (12.39) shows that in general, a wave plate will only be a quarter wave plate for light of a definite frequency.

You can check that the matrices are \[P_{\pm}=\frac{1}{2}\left(\begin{array}{cc} 1 & \mp i \\ \pm i & 1 \end{array}\right) .\]

Reflection occurs at interfaces between materials of different indices of refraction. The larger the difference in refractive index the greater the reflectivity. At wavelengths < 50 nm, all materials have indices of refraction ≈ 1. Thus, it is difficult to create a highly reflective interface. At EUV wavelengths, it has proven possible to make mirrors with moderate reflectivity, in the range of 60–70%, by the use of multilayers. Multilayer reflectors are made by depositing alternating layers of high-Z and low-Z materials, giving a small but effective difference between refractive indices at each interface (Fig. 12.9). By making the periodicity d of the multilayer stack satisfy the Bragg condition,

where the phases are random and the frequencies are chosen at random in a small range around a central frequency. You can watch the \(\vec{E}\) field wandering in the \(x\)-\(y\) plane, eventually filling it up. The narrower the range of frequencies in the wave, the more slowly the polarization wanders. In the example in program 12-2, the range of frequencies is of the order of 10% of the central frequency, so the polarization wanders rapidly. But for a beam with a fairly well-defined frequency, the polarization will be nearly constant over many cycles of the wave. The time over which the polarization is approximately constant is called the coherence time of the wave. For a plane wave of definite frequency, the coherence time is infinite.

“Optical activity” is a property of many organic and some inorganic compounds. An optically active material rotates the polarization of light without absorbing either component of the polarization. A familiar example of such a material is corn syrup, a thick aqueous solution of sugar that you probably have in your kitchen. If you put a rectangular container of corn syrup between polarizers, as shown in Figure \( 12.7\), and rotate the second polarizer until the intensity of the light getting through is a maximum, you will find that direction of the second polarizer is not the same as that of the first. The plane of the polarization has been rotated by some angle \(\theta\). The rotation angle, \(\theta\), is proportional to the thickness of the container, the length of the region of syrup that the light goes through.

The effects of wave plates and polarizers and the like can be summarized by multiplication of the \(Z\) vector by 2×2 matrices. For example, a perfect polarizer with an axis at an angle \(\theta\) from the 1 axis can be represented by \[P_{\theta}=\left(\begin{array}{cc} \cos ^{2} \theta & \cos \theta \sin \theta \\ \cos \theta \sin \theta & \sin ^{2} \theta \end{array}\right) .\]

The very short wavelength enables a small numerical aperture to be used, avoiding the problem of self-vignetting in all-reflective optics. Wavelengths in the range 11–14 nm are in the extreme ultraviolet (EUV) or soft x-ray portion of the electromagnetic spectrum, so lithography using such wavelengths is referred to as EUV lithography. Because projection optics are possible at EUV wavelengths, particularly reduction optics, many of the mask problems encountered with 1:1 x-ray lithography are avoided. However, as will be seen, EUV lithography has its own set of challenges.

This page titled 12.3: Wave Plates and Polarizers is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Howard Georgi via source content that was edited to the style and standards of the LibreTexts platform.

You can find a clue to the nature of optical activity by considering what it looks like if you look at it in a mirror. If you reflect the system illustrated in Figure \( 12.7\) in the \(x\)-\(z\) plane, by changing the sign of all the \(y\) coordinates, the angle \(\theta\) changes to −\(\theta\). Thus the corn syrup that you see in a mirror must be fundamentally different from the corn syrup in your kitchen. This is not so strange. After all, your right hand looks like a left hand when you look at it in a mirror. The corn syrup must have the same property and have a definite “handedness.” In fact, because of the tetrahedral bonding of the carbon atoms of which they are built, the sugar molecules in the corn syrup can and do have such a handedness.

In the opposite order, \(P_{\pi / 4} Q_{0}\) is an analyzer for circularly polarized light. It annihilates counterclockwise light and converts clockwise polarized light to light linearly polarized in the \(\pi / 4\) direction.

Reflectance versus wavelength is shown in Fig. 12.10. As can be seen in this figure, reflectance peaks narrowly around a given wavelength. From Eq. (12.8), it is apparent that only a very small wavelength shift causes a small change in peak wavelength (see Problem 12.3.) Even if every mirror in the EUV system has very high peak reflectances, the overall system transmission can be low if these peak reflectances do not occur at nearly the same wavelength. A similar statement is true regarding the need to match the peak reflectance of the masks to that of the projection optics mirrors.4 When specifying the wavelength at which peak reflectance occurs, it is important to pay attention to detail. As can be seen in Fig. 12.10, the curve of reflectance versus wavelength is asymmetric.

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as it must, since the first polarizer produces polarized light and the second one transmits it perfectly. \(P_{\theta}\) acting on a vector transmits the component in the \(\theta\) direction. This is easiest to visualize if \(\theta = 0\) or \(\pi / 2\). The matrices \[P_{0}=\left(\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right), \quad P_{\pi / 2}=\left(\begin{array}{ll} 0 & 0 \\ 0 & 1 \end{array}\right) ,\]

A wave plate in which the phase difference is \(\pi\) is called a “half wave plate.” A half wave plate is obtained by replacing the \(i\) in (12.45)-(12.47) by -1. Thus, \[H_{\theta}=P_{\theta}-P_{\theta+\pi / 2} .\]

As discussed in Chapter 5, optical lithography has been practiced where there are light sources that satisfy certain key requirements, particularly narrow bandwidth and high intensity. The optics and masks of each optical lithographic technology have been engineered around this primary requirement of an intense, narrow bandwidth light source. For EUV lithography there are few options for masks and optics. EUV lithography must necessarily be practiced at wavelengths where there are multilayer reflectors with at least moderate reflectance. Unfortunately, intense sources of light may not exist at these wavelengths. Productivity due to low light intensity at the wafer is a concern with EUV lithographic technology, so considerable attention must be paid to multilayer reflectance and sources of EUV light.

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Polarization offers many opportunities to get confused when you think of the light wave in terms of photons. Let us imagine turning down the intensity of the light to the point where one photon at a time is going through the polarizers and consider first the deceptively simple situation of light moving in the \(z\) direction through crossed polarizers in the \(x\)-\(y\) plane. Suppose that the first polarizer transmits light polarized in the \(x\) direction, and the second transmits light polarized in the \(y\) direction. This is deceptively simple because it seems that we can interpret what is going on simply in terms of photons. The situation is depicted in Figure \( 12.8\). This seems simple enough to interpret in terms of photons. The unpolarized light in region \(I\) is composed equally of photons polarized in the \(x\) direction and in the \(y\) direction (goes the wrong “classical” argument). Those polarized in the \(x\) direction get through the first polarizer, so half the photons are still around in region \(II\), where the intensity is reduced by half. Then none of these get through the second polarizer, so that the intensity in region \(III\) is zero.

Coming out of the first polarizer, the vector, \(Z\), looks like (12.40) for all the frequency components in the white light. But when the wave plate is inserted in between, a frequency dependent phase difference is added, so that the \(Z\) vector coming out of the wave plate (up to an irrelevant overall phase) looks like \[Z=\left(\begin{array}{c} 1 / \sqrt{2} \\ e^{-i \Delta \phi} / \sqrt{2} \end{array}\right) .\]

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Because of the handedness of the sugar molecules, the index of refraction of the corn syrup actually depends on the handedness of the light. It is slightly different for left- and right-circularly polarized light. This happens because the \(\vec{E}\) field of a circularly polarized beam twists slightly as it traverses each sugar molecule and sees a slightly different electronic structure depending on the direction of the twist. Then, because the indices of refraction are slightly different, the left- and right-circularly polarized components of the light get different phase factors (\(k \ell\)) in passing through a thickness, \(\ell\), of the syrup.

When this acts on an arbitrary vector you get circularly polarization unless the vector is annihilated by \(P_{\pi / 4}\). \[Q_{0} P_{\pi / 4}\left(\begin{array}{l} \psi_{1} \\ \psi_{2} \end{array}\right)=\frac{\psi_{1}+\psi_{2}}{2}\left(\begin{array}{l} 1 \\ i \end{array}\right) .\]

The materials chosen to comprise the multilayer stack must be weak absorbers of EUV light, since the light must be able to penetrate to the lower layers of the film stack. Several stacks have been identified as possible reflectors in the EUV, and these are listed in Table 12.1. Because of beryllium’s toxicity, there is a reluctance to use that metal for EUV lithography, and most effort in EUV lithography today is focused on Mo/Si reflectors. Because reflectance is angle dependent [Eq. (12.5)], graded depositions are needed on optics that have light incident at varying angles over the mirror surfaces.

It is this frequency dependence that produces the interesting patterns of color that you see when you put cellophane or a stressed piece of plastic between polarizers.

Then (12.52) becomes \[e^{-i \theta} \frac{1}{2}\left(\begin{array}{cc} 1 & -i \\ i & 1 \end{array}\right)+e^{i \theta} \frac{1}{2}\left(\begin{array}{cc} 1 & i \\ -i & 1 \end{array}\right)=\left(\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right) .\]

For example, the transparent polymer material cellophane is made into thin sheets by stretching. Because of the stretching, the polymer strands tend to be oriented along the stretch direction. The dielectric constant in this material depends on the direction of the electric field. It is easier for charges to move along the polymer strands than across them. Thus the dielectric constant is larger for electric fields in the stretch direction.

where θ is defined in Fig. 12.9, the net effect of small reflectivity at each interface is moderately high reflectivity overall when the stack has a sufficient number of layers.