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Suppose \(n_{e}>n_{o}\) and that the fast axis, which corresponds to \(n_{o}\) is aligned with \(\mathcal{E}_{x}\), while the slow axis (which then has refractive index \(n_{e}\) ) is aligned with \(\mathcal{E}_{y}\). If the wave travels a distance \(d\) through the crystal, \(\mathcal{E}_{y}\) will accumulate a phase \(\Delta \varphi_{y}=\frac{2 \pi n_{e}}{\lambda} d\), and \(\mathcal{E}_{x}\) will accumulate a phase \(\Delta \varphi_{x}=\frac{2 \pi n_{o}}{\lambda} d\). Thus, after propagation through the crystal the phase difference \(\varphi_{y}-\varphi_{x}\) has increased by \[\Delta \varphi_{y}-\Delta \varphi_{x}=\frac{2 \pi}{\lambda} d\left(n_{e}-n_{o}\right) . \nonumber \]

One of the advantages of IR observation is that it can detect objects that are too cool to emit visible light. This has led to the discovery of previously unknown objects, including comets, asteroids and wispy interstellar dust clouds that seem to be prevalent throughout the galaxy.

Caltech describes infrared astronomy as "the detection and study of the infrared radiation (heat energy) emitted from objects in the universe." Advances in IR CCD imaging systems have allowed for detailed observation of the distribution of IR sources in space, revealing complex structures in nebulas, galaxies and the large-scale structure of the universe.

That \(\mathcal{R}(\theta)\) indeed is a rotation over angle \(\theta\) in the anti-clockwise direction is easy to see by considering what happens when \(\mathcal{R}_{\theta}\) is applied to the vector \((1,0)^{T}\). Since \(\mathcal{R}_{\theta}^{-1}=\mathcal{R}_{-\theta}\) we get: \[\left(\begin{array}{c} E_{x^{\prime}} \\ E_{y^{\prime}} \end{array}\right)=\mathcal{R}_{-\theta}\left(\begin{array}{c} E_{x} \\ E_{y} \end{array}\right) . \nonumber \]

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Infrared radiation (IR), or infrared light, is a type of radiant energy that's invisible to human eyes but that we can feel as heat. All objects in the universe emit some level of IR radiation, but two of the most obvious sources are the sun and fire.

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So suppose that with one of these methods we have obtained linearly polarised light. Then the question rises how the state of linear polarisation can be changed into circularly or elliptically polarised light. Or how the state of linear polarisation can be rotated over a certain angle. We have seen that the polarisation state depends on the ratio of the amplitudes and on the phase difference \(\varphi_{y}-\varphi_{x}\) of the orthogonal components \(\mathcal{E}_{y}\) and \(\mathcal{E}_{x}\) of the electric field. Thus, to change linearly polarised light to some other state of polarisation, a certain phase shift (say \(\Delta \varphi_{x}\) ) must be introduced to one component (say \(\mathcal{E}_{x}\) ), and another phase shift \(\Delta \varphi_{y}\) to the orthogonal component \(\mathcal{E}_{y}\). We can achieve this with a birefringent crystal, such as calcite. What is special about such a crystal is that it has two refractive indices: light polarised in a certain direction experiences a refractive index of \(n_{o}\), while light polarised perpendicular to it feels another refractive index \(n_{e}\) (the subscripts \(o\) and \(e\) stand for "ordinary" and "extraordinary"), but for our purpose we do not need to understand this terminology. The direction for which the refractive index is smallest (which can be either \(n_{o}\) or \(n_{e}\) ) is called the fast axis because its phase velocity is largest, and the other direction is the slow axis. Because there are two different refractive indices, one can see double images through a birefringent crystal. The difference between the two refractive indices \(\Delta n=n_{e}-n_{o}\) is called the birefringence.

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Light that is a mixture of polarised and unpolarised light is called partially polarised. The degree of polarisation is defined as the fraction of the total intensity that is polarised: \[\text { degree of polarisation }=\frac{I_{p o l}}{I_{p o l}+I_{\text {unpol }}} \text {. } \nonumber \]

We want to change it to circularly polarised light, for which \[J=\frac{1}{\sqrt{2}}\left(\begin{array}{l} 1 \\ i \end{array}\right) \text {, } \nonumber \] where one can check that indeed \(\varphi_{y}-\varphi_{x}=\pi / 2\). This can be done by passing the light through a crystal such that \(\mathcal{E}_{y}\) accumulates a phase difference of \(\pi / 2\) with respect to \(\mathcal{E}_{x}\). The transformation by which this is accomplished can be written as \[\left(\begin{array}{ll} 1 & 0 \\ 0 & i \end{array}\right) \frac{1}{\sqrt{2}}\left(\begin{array}{l} 1 \\ 1 \end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{l} 1 \\ i \end{array}\right) . \nonumber \]

In particular, if incident light is linear polarised under \(45^{\circ}\), or equivalently, if the quarter wave plate is rotated over this angle, it will transform linearly polarised light into circularly polarised light (and vice versa). \[\frac{1}{\sqrt{2}}\left(\begin{array}{l} 1 \\ i \end{array}\right)=\left(\begin{array}{ll} 1 & 0 \\ 0 & i \end{array}\right) \frac{1}{\sqrt{2}}\left(\begin{array}{l} 1 \\ 1 \end{array}\right) \nonumber \]

Natural light often does not have a definite polarisation. Instead, the polarisation fluctuates rapidly with time. In order to turn such randomly polarised light into linearly polarised light in a certain direction, we must extinguish the light polarised in the perpendicular direction, so that the remaining light is linearly polarised along the required direction. One could do this by using light reflected under the Brewster angle (which extinguishes p-polarised light), or one could let light pass through a dichroic crystal, which is a material which absorbs light polarised perpendicular to its so-called optic axis. A third method is sending the light through a wire grid polariser, which consists of a metallic grating with sub-wavelength slits. Such a grating only transmits the electric field component that is perpendicular to the slits.

Comparing with ( \(\PageIndex{7}\) ) implies \[\left(\begin{array}{l} E_{x} \\ E_{y} \end{array}\right)=\left(\begin{array}{l} E_{x^{\prime}} \cos \theta-E_{y^{\prime}} \sin \theta \\ E_{x^{\prime}} \sin \theta+E_{y^{\prime}} \cos \theta \end{array}\right)=\mathcal{R}_{\theta}\left(\begin{array}{c} E_{x^{\prime}} \\ E_{y^{\prime}} \end{array}\right), \nonumber \] where \(\mathcal{R}_{\theta}\) is the rotation matrix over an angle \(\theta\) in the anti-clockwise direction: \[\mathcal{R}_{\theta} \equiv\left(\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right) \nonumber \]

A simple example of such a sensor is the bolometer, which consists of a telescope with a temperature-sensitive resistor, or thermistor, at its focal point, according to the University of California, Berkeley (UCB). If a warm body comes into this instrument's field of view, the heat causes a detectable change in the voltage across the thermistor.

IR radiation is one of the three ways heat is transferred from one place to another, the other two being convection and conduction. Everything with a temperature above around 5 degrees Kelvin (minus 450 degrees Fahrenheit or minus 268 degrees Celsius) emits IR radiation. The sun gives off half of its total energy as IR, and much of the star's visible light is absorbed and re-emitted as IR, according to the University of Tennessee.

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By letting light pass through crystals of different thicknesses \(d\), we can create different phase differences between the orthogonal field components, and this way we can create different states of polarisation. To be specific, let \(\mathbf{J}\), as given by (4.1.4), be the Jones vector of the plane wave before the crystal. Then we have, for the Jones vector after the passage through the crystal: \[\tilde{\mathbf{J}}=\mathcal{M} \mathbf{J}, \nonumber \] where \[\mathcal{M}=\left(\begin{array}{cc} e^{\frac{2 \pi i}{\lambda} d n_{o}} & 0 \\ 0 & e^{\frac{2 \pi i}{\lambda} d n_{e}} \end{array}\right)=e^{\frac{2 \pi i}{\lambda} d n_{o}}\left(\begin{array}{cc} 1 & 0 \\ 0 & e^{\frac{2 \pi i}{\lambda} d\left(n_{e}-n_{o}\right)} \end{array}\right) . \nonumber \] A matrix such as \(\mathcal{M}\), which transfers one state of polarisation of a plane wave in another, is called a Jones matrix. Depending on the phase difference which a wave accumulates by traveling through the crystal, these devices are called quarter-wave plates (phase difference \(\pi / 2\) ), half-wave plates (phase difference \(\pi\) ), or full-wave plates (phase difference \(2 \pi\) ). The applications of these wave plates will be discussed in later sections.

We have seen how Maxwell’s equations allow the existence of plane waves with many different states of polarisation. But how can we create these states, and how do these states manifest themselves?

Let \(\mathbf{E}\) be given in terms of its components on the \(\hat{\mathbf{x}}, \hat{\mathbf{y}}\) basis: \[\mathbf{E}=E_{x} \widehat{\mathbf{x}}+E_{y} \widehat{\mathbf{y}} . \nonumber \]

Natural light such as sun light is unpolarised. The instantaneous polarisation of unpolarised light fluctuates rapidly in a random manner. A linear polariser produces linear polarised light from unpolarised light.

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Household appliances such as heat lamps and toasters use IR radiation to transmit heat, as do industrial heaters such as those used for drying and curing materials. Incandescent bulbs convert only about 10 percent of their electrical energy input into visible light energy, while the other 90 percent is converted to infrared radiation, according to the Environmental Protection Agency.

Infrared spectroscopy measures IR emissions from materials at specific wavelengths. The IR spectrum of a substance will show characteristic dips and peaks as photons (particles of light) are absorbed or emitted by electrons in molecules as the electrons transition between orbits, or energy levels. This spectroscopic information can then be used to identify substances and monitor chemical reactions.

A half-wave plate introduces a phase shift of \(\pi\), so its Jones matrix is \[\mathcal{M}_{H W P}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right), \nonumber \] because \(\exp (i \pi)=-1\). An important application of the half-wave plate is to change the orientation of linearly polarised light. After all, what this matrix does is mirroring the polarisation state in the \(x\)-axis. Thus, if we choose our mirroring axis correctly (i.e. if we choose the orientation of the wave plate correctly), we can change the direction in which the light is linearly polarised arbitrarily. A demonstration is shown in. To give an example: the polarisation of a wave that is parallel to the \(x\)-direction, can be rotated over angle \(\alpha\) by rotating the crystal such that the slow axis makes angle \(\alpha / 2\) with the \(x\)-axis. Upon propagation through the crystal, the fast axis gets an additional phase of \(\pi\), due to which the electric vector makes angle \(\alpha\) with the \(x\)-axis (see Figure \(\PageIndex{2}\) ).

According to Robert Mayanovic, professor of physics at Missouri State University, infrared spectroscopy, such as Fourier transform infrared (FTIR) spectroscopy, is highly useful for numerous scientific applications. These include the study of molecular systems and 2D materials, such as graphene.

Stateof polarization of light

Consider as example the Jones matrix which described the change of linear polarised light into circular polarisation. Assume that we have diagonally (linearly) polarised light, so that \[J=\frac{1}{\sqrt{2}}\left(\begin{array}{l} 1 \\ 1 \end{array}\right) \text {. } \nonumber \]

Clearly, horizontally polarised light is completely transmitted, while vertically polarised light is not transmitted at all. More generally, for light that is polarised at an angle \(\alpha\), we get \[\mathcal{M}_{\alpha}=\mathcal{M}_{L P}\left(\begin{array}{c} \cos \alpha \\ \sin \alpha \end{array}\right)=\left(\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right)\left(\begin{array}{c} \cos \alpha \\ \sin \alpha \end{array}\right)=\left(\begin{array}{c} \cos \alpha \\ 0 \end{array}\right) \text {. } \nonumber \]

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A full-wave plate introduces a phase difference of \(2 \pi\), which is the same as introducing no phase difference between the two field components. So what can possibly be an application for a full-wave plate? We need to recall from Eq. (( \(\PageIndex{1}\) )) that the phase difference is \(2 \pi\) only for a particular wavelength. If we send through linearly (say vertically) polarised light of other wavelengths, these will become elliptically polarised, while the light with the correct wavelength \(\lambda_{0}\) will stay vertically polarised. If we then let all the light pass through a horizontal polariser, the light with wavelength \(\lambda_{0}\) will be completely extinguished, while the light of other wavelengths will be able to pass through at least partially. Therefore, full-wave plates can be used to filter out specific wavelengths of light.

To find the components \(E_{x^{\prime}}, E_{y^{\prime}}\) on the \(\widehat{\mathbf{x}}^{\prime}, \widehat{\mathbf{y}}^{\prime}\) basis: \[\mathbf{E}=E_{x^{\prime}} \widehat{\mathbf{x}}^{\prime}+E_{y^{\prime}} \widehat{\mathbf{y}}^{\prime} , \nonumber \] we first write the unit vectors \(\widehat{\mathbf{x}}^{\prime}\) and \(\widehat{\mathbf{y}}^{\prime}\) in terms of the basis \(\hat{\mathbf{x}}, \hat{\mathbf{y}}\) (see Figure \(\PageIndex{1}\) ) \[\begin{aligned} &\widehat{\mathbf{x}}^{\prime}=\cos \theta \widehat{\mathbf{x}}+\sin \theta \widehat{\mathbf{y}}, \\ &\widehat{\mathbf{y}}^{\prime}=-\sin \theta \widehat{\mathbf{x}}+\cos \theta \widehat{\mathbf{y}}.\end{aligned} \nonumber \]

The amplitude of the transmitted field is reduced by the factor \(\cos \alpha\), which implies that the intensity of the transmitted light is reduced by the factor \(\cos ^{2} \alpha\). This relation is known as Malus’ law.

Night vision cameras use a more sophisticated version of a bolometer. These cameras typically contain charge-coupled device (CCD) imaging chips that are sensitive to IR light. The image formed by the CCD can then be reproduced in visible light. These systems can be made small enough to be used in hand-held devices or wearable night-vision goggles. The cameras can also be used for gun sights with or without the addition of an IR laser for targeting.

IR is a type of electromagnetic radiation, a continuum of frequencies produced when atoms absorb and then release energy. From highest to lowest frequency, electromagnetic radiation includes gamma-rays, X-rays, ultraviolet radiation, visible light, infrared radiation, microwaves and radio waves. Together, these types of radiation make up the electromagnetic spectrum.

It follows from ( \(\PageIndex{17}\) ) that the intensity transmitted by a linear polariser when unpolarised light is passed incident, is the average value of \(\cos ^{2} \alpha\) namely \(\frac{1}{2}\), times the incident intensity.

British astronomer William Herschel discovered infrared light in 1800, according to NASA. In an experiment to measure the difference in temperature between the colors in the visible spectrum, he placed thermometers in the path of light within each color of the visible spectrum. He observed an increase in temperature from blue to red, and he found an even warmer temperature measurement just beyond the red end of the visible spectrum.

A polariser that only transmits horizontally polarised light is described by the Jones matrix: \[\mathcal{M}_{L P}=\left(\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right) . \nonumber \]

This relationship expresses the components \(E_{x^{\prime}}, E_{y^{\prime}}\) of the Jones vector on the \(\hat{\mathbf{x}}^{\prime}, \widehat{\mathbf{y}}^{\prime}\) basis, which is aligned with the fast and slow axes of the crystal, in terms of the components \(E_{x}\) and \(E_{y}\) on the original basis \(\widehat{\mathbf{x}}, \widehat{\mathbf{y}}\). If the matrix \(\mathcal{M}\) describes the Jones matrix as defined in ( \(\PageIndex{3}\) ), then the matrix \(M_{\theta}\) for the same wave plate but with \(x^{\prime}\) as slow and \(y^{\prime}\) as fast axis, is, with respect to the \(\widehat{\mathbf{x}}, \widehat{\mathbf{y}}\) basis, given by: \[\mathcal{M}_{\theta}=\mathcal{R}_{\theta} \mathcal{M} \mathcal{R}_{-\theta} . \nonumber \]

Another advantage of IR radiation is that its longer wavelength means it doesn't scatter as much as visible light, according to NASA. Whereas visible light can be absorbed or reflected by gas and dust particles, the longer IR waves simply go around these small obstructions. Because of this property, IR can be used to observe objects whose light is obscured by gas and dust. Such objects include newly forming stars imbedded in nebulas or the center of Earth's galaxy.

Similar to the visible light spectrum, which ranges from violet (the shortest visible-light wavelength) to red (longest wavelength), infrared radiation has its own range of wavelengths. The shorter "near-infrared" waves, which are closer to visible light on the electromagnetic spectrum, don't emit any detectable heat and are what's discharged from a TV remote control to change the channels. The longer "far-infrared" waves, which are closer to the microwave section on the electromagnetic spectrum, can be felt as intense heat, such as the heat from sunlight or fire, according to NASA.

IR astronomy is particularly useful for observing cold molecules of gas and for determining the chemical makeup of dust particles in the interstellar medium, said Robert Patterson, professor of astronomy at Missouri State University. These observations are conducted using specialized CCD detectors that are sensitive to IR photons.

A quarter-wave plate introduces a phase shift of \(\pi / 2\), so its Jones matrix is \[\mathcal{M}_{Q W P}=\left(\begin{array}{ll} 1 & 0 \\ 0 & i \end{array}\right), \nonumber \] because \(\exp (i \pi / 2)=i\). To describe the actual transmission through the quarter-wave plate, the matrix should be multiplied by some global phase factor, but because we only care about the phase difference between the field components, this global phase factor can be omitted without problem. The quarter-wave plate is typically used to convert linearly polarised light to elliptically polarised light and vice-versa. If the incident light is linearly polarised at angle \(\alpha\), the state of polarisation after the quater wave plate is \[\left(\begin{array}{c} \cos \alpha \\ i \sin \alpha \end{array}\right)=\left(\begin{array}{ll} 1 & 0 \\ 0 & i \end{array}\right)\left(\begin{array}{c} \cos \alpha \\ \sin \alpha \end{array}\right) \text {. } \nonumber \]

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By substituting ( \(\PageIndex{9}\) ) and ( \(\PageIndex{10}\) ) into ( \(\PageIndex{8}\) ) we find \[\begin{aligned} \mathbf{E} &=E_{x^{\prime}} \widehat{\mathbf{x}}^{\prime}+E_{y^{\prime}} \widehat{\mathbf{y}}^{\prime} \\ &=E_{x^{\prime}}(\cos \theta \widehat{\mathbf{x}}+\sin \theta \widehat{\mathbf{y}})+E_{y^{\prime}}(-\sin \theta \widehat{\mathbf{x}}+\cos \theta \widehat{\mathbf{y}}), \\ &=\left(\cos \theta E_{x^{\prime}}-\sin \theta E_{y^{\prime}}\right) \widehat{\mathbf{x}}+\left(\sin \theta E_{x}+\cos \theta E_{y}\right) \widehat{\mathbf{y}} . \end{aligned} \nonumber \]

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Within the electromagnetic spectrum, infrared waves occur at frequencies above those of microwaves and just below those of red visible light, hence the name "infrared." Waves of infrared radiation are longer than those of visible light, according to the California Institute of Technology (Caltech). IR frequencies range from about 300 gigahertz (GHz) up to about 400 terahertz (THz), and wavelengths are estimated to range between 1,000 micrometers (µm) and 760 nanometers (2.9921 inches), although these values are not definitive, according to NASA.

4.3: Creating and Manipulating Polarisation States is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

Another important Jones matrix is the rotation matrix. In the preceding discussion it was assumed that the fast and slow axes were aligned with the \(x\) - and \(y\)-direction (i.e. they were parallel to \(\mathcal{E}_{x}\) and \(\mathcal{E}_{y}\) ). Suppose now that the slow and fast axes of the wave plate no longer coincide with \(\widehat{\mathbf{x}}\) and \(\widehat{\mathbf{y}}\), but rather with some other \(\widehat{\mathbf{x}}^{\prime}\) and \(\widehat{\mathbf{y}}^{\prime}\) as in Figure \(\PageIndex{1}\). In that case we apply a basis transformation: the electric field vector which is expressed in the \(\widehat{\mathbf{x}}, \widehat{\mathbf{y}}\) basis should first be expressed in the \(\widehat{\mathbf{x}}^{\prime}, \widehat{\mathbf{y}}^{\prime}\) basis before applying the Jones matrix of the wave plate to it. After applying the Jones matrix, the electric field has to be transformed back from the \(\widehat{\mathbf{x}}^{\prime}, \widehat{\mathbf{y}}^{\prime}\) basis to the \(\widehat{\mathbf{x}}, \widehat{\mathbf{y}}\) basis.

Infrared lasers can be used for point-to-point communications over distances of a few hundred meters or yards. TV remote controls that rely on infrared radiation shoot out pulses of IR energy from a light-emitting diode (LED) to an IR receiver in the TV, according to How Stuff Works. The receiver converts the light pulses to electrical signals that instruct a microprocessor to carry out the programmed command.

One of the most useful applications of the IR spectrum is in sensing and detection. All objects on Earth emit IR radiation in the form of heat. This can be detected by electronic sensors, such as those used in night vision goggles and infrared cameras.