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.

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 \[\left(\begin{array}{ll} 1 & 0 \\ 0 & i \end{array}\right) \frac{1}{\sqrt{2}}\left(\begin{array}{l} 1 \\ 1 \end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{l} 1 \\ i \end{array}\right) . \nonumber \]

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?

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). \[\frac{1}{\sqrt{2}}\left(\begin{array}{l} 1 \\ i \end{array}\right)=\left(\begin{array}{ll} 1 & 0 \\ 0 & i \end{array}\right) \frac{1}{\sqrt{2}}\left(\begin{array}{l} 1 \\ 1 \end{array}\right) \nonumber \]

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.

s-polarization vs p polarization

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 \(\PageIndex{1}\) ) \[\begin{aligned} &\widehat{\mathbf{x}}^{\prime}=\cos \theta \widehat{\mathbf{x}}+\sin \theta \widehat{\mathbf{y}}, \\ &\widehat{\mathbf{y}}^{\prime}=-\sin \theta \widehat{\mathbf{x}}+\cos \theta \widehat{\mathbf{y}}.\end{aligned} \nonumber \]

What is polarization in Chemistry

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: \[\left(\begin{array}{c} E_{x^{\prime}} \\ E_{y^{\prime}} \end{array}\right)=\mathcal{R}_{-\theta}\left(\begin{array}{c} E_{x} \\ E_{y} \end{array}\right) . \nonumber \]

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: \[\mathcal{M}_{\theta}=\mathcal{R}_{\theta} \mathcal{M} \mathcal{R}_{-\theta} . \nonumber \]

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Stateof polarization of light

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

This white paper thoroughly discusses single mode optical fiber, multimode optical fiber, the difference between both, and the selection parameters for specific operations.

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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.

The mode is generally defined as the way of optical wave transmission. The optical waves of uniform frequencies, as they enter in the optical fiber are distributed in different paths of transmission. The path followed by each individual optical wave is referred to as mode.

Polarisation statemeaning

Single mode optical fiber is a type of optical fiber designed specifically for single mode light transmission. This means different light waves of different frequencies are transmitted through a single path via this type of optical fiber. These single mode optical fibers are preferred in the industry by a common acronym SMF. The SMF carries optical signals in a transverse mode. That means the path of optic wave transmission is parallel to the length of fiber optic core strand but the electromagnetic oscillation takes place in a perpendicular/transverse direction.

Depending on the variation in characteristics, the single mode optical fiber (SMF) is categorized. Let us discuss the categorization of SMF further.

Circularpolarisation state

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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.

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

An optical fiber or fiber optics has emerged as a leading signal transmission medium in data networking applications. It was first proposed as a technology for light transmission in the 1950s. By 1990s the optical fiber technology has gained traction. By 2010, it has taken over the copper cable transmission. Unlike, copper cables, the fiber optics transmits the signals in an optic format that is in the form of light waves. It offers higher data transmission speed, greater data security, and endurance to harsh environmental conditions in comparison to the copper cables. These cables benefit from their design and construction. The fiber optic cables are constructed by bundling multiple glass strands inside a cladding, which is further protected by a buffer tube, which offers protection against environmental conditions. The optic signals are transmitted through these glass strands by the refraction of light principle.

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

Besides the characteristics, the differences between the single mode and multimode optical fiber can be stated in a few different ways. The following are the highlighted differences between single mode and multimode optical fibers.

The optic signals follow a path of transmission inside the glass strand, which is known as the mode of transmission. In short, the mode of transmission refers to optic signal transmission from one end of fiber optic cable to another. The mode of transmission depends on the construction, type of glass material, refractive index, cladding efficiency, and the inner diameter of the transmission media that is the glass strand. There are two fundamental types of fiber optics cables available based on the mode of transmission, namely, single mode and multimode fiber optic cables. These fiber optic cables are often utilized for various applications under different conditions. Therefore, it is essential to know the selection parameters of single mode or multimode optical fiber specific applications.

Due to the extensively growing demand for fiber optic technology, numerous applications of fiber optic cables are found. However, due to the versatility of applications and requirements, the selection of single mode or multimode fiber optic cable has become crucial. Owing to the need for efficiency, it is essential to consider several factors related to the application and fiber optic in order to make the right choice. Thorough knowledge about fiber optic technology, single mode optical fiber and multimode optical fiber, their characteristics, and scope of utility is important, this can certainly lead to an effective selection of suitable fiber optic cable. To gain more information about VERSITRON's SMF and MMF cables and related equipment, please contact us.

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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The number of modes in a fiber optic cable is calculated by solving the Helmholtz equation for wave. The Helmholtz equation itself is obtained by applying boundary conditions to Maxwell’s equation. Therefore, the fiber optic modes are projectile solutions of Maxwell’s equation.

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 \[\mathcal{M}_{\alpha}=\mathcal{M}_{L P}\left(\begin{array}{c} \cos \alpha \\ \sin \alpha \end{array}\right)=\left(\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right)\left(\begin{array}{c} \cos \alpha \\ \sin \alpha \end{array}\right)=\left(\begin{array}{c} \cos \alpha \\ 0 \end{array}\right) \text {. } \nonumber \]

Polarizedstateof neuron

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.

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 \(\PageIndex{2}\) ).

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: \[\mathcal{R}_{\theta} \equiv\left(\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right) \nonumber \]

When it comes to select between single mode and multimode fiber optic for a specific application, one must consider several factors. The factors of consideration are an application requirement, cost of fiber, cost of installation system, equipment installation requirements, a distance of transmission, speed of transmission, etc. In order to select one of the single mode or multimode optical fibers, the thorough comparison for these selection factors shall be done.Refer to the following comparison table for convenient selection.

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.

The multimode optical fiber (MMF) is mainly categorized into two types based on two factors, namely, a system of classification and refractive index and signal behavior.

Polarisationmeaning in Physics

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

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 \[\Delta \varphi_{y}-\Delta \varphi_{x}=\frac{2 \pi}{\lambda} d\left(n_{e}-n_{o}\right) . \nonumber \]

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: \[\text { degree of polarisation }=\frac{I_{p o l}}{I_{p o l}+I_{\text {unpol }}} \text {. } \nonumber \]

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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 \[\left(\begin{array}{c} \cos \alpha \\ i \sin \alpha \end{array}\right)=\left(\begin{array}{ll} 1 & 0 \\ 0 & i \end{array}\right)\left(\begin{array}{c} \cos \alpha \\ \sin \alpha \end{array}\right) \text {. } \nonumber \]

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.

Polarisation stateexample

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The multimode optical fiber is a type of optical fiber designed for multiple light signal propagation. The industrial acronym for multimode optical fiber is MMF. In MMF, the modal dispersion takes place according to varying wavelengths of the optical signals. Therefore, modal dispersion in the MMF is higher. The path of optic wave propagation in the MMF is either zigzag or semi-elliptical in nature, it depends on the refractive index of the glass core material.

The information provided in this white paper is intended solely for general information purposes. The practice of Engineering differs across each project, as it is driven by site-specific circumstances. Thus, any business decision based on the implementation must be taken only after consultation with a qualified and licensed professional who is capable of addressing all relevant factors, challenges, and desired outcomes. The information in these white papers is derived from various verified sources and posted after reasonable care and attention. It is possible that some information may appear incomplete, incorrect, or inapplicable considering your particular condition. In such condition, VERSITRON does not accept the liability for direct or indirect losses resulting from using, relying, or acting upon the information in this white paper.

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The modes are the result of the modal dispersion phenomenon taking place inside the fiber optic cable.Note: The modal dispersion is totally irrelevant of the number of glass fiber strands wrapped inside the cladding.