Light, polarized

According to Maxwell s equations, light is a traveling electromagnetic wave for our purposes, only the electric field component is of importance. For a beam propagating in the z direction, as will be assumed henceforth, the electric field amplitude E oscillates in the x-y plane, and the wave may be expressed in either of two equivalent ways  

Schematic illustration of (a) y-polarized, (b) Jc-polaiiz and (c) un-polariz light. 

Light waves obey the principle of superposition, and thus the net effect of the two waves can be conq)uted simply by adding eqs. 9.3.3 and 9.3.4 together. (Conversely, any given wave can be viewed as an appropriate sum of components along selected axes.) This sum depends critically on p, as illustrated by the following specific cases. 

Here Ey has its positive and negative maxima when Ey = 0, and vice versa. Note what happens when l jrl = o/(2) , which 

Given that the birefringence An is a property of the material, it remains ordy to cut a thickness L such that i — njl. The quantity h is called the retardation, for obvious reasons aquarter-wave plate retards one component by n/1 rad, or one-quarter of a wave, relative to the other. Note that in a birefringence experiment, it is always a retardation that is measured directly, not the birefringence itself. 

The various forms of polarization can be understood by considering a plane-polarized wave travelling in the z direction. It can be resolved into two 

Confining attention to the electric fields, we have for the x wave, 

The above treatment is general, but the following special cases are important. 

The tip of E describes a straight line passing through the origin with slope 

The paths described by the tip of E for the various values of 5 are summarized in Fig. 8.3. It is apparent that plane-polarized light is a special case of elliptically polarized light, as indeed is circularly polarized light. The sense of the 

The light generated by common sources (sun, lamps) is composed of an ensemble of waves whose polarization planes are randomly oriented around the travelling direction. A radiation of this type is said to be unpolarized. Conversely, if all the waves that constitute the radiation possess the same polarization plane, the wave is linearly polarized (Fig. 6.1). The oscillation direction of the electric field is called polarization direction—or simply polarization—of the wave. 

Circular polarization of a wave can occur in two directions right or left. For a right circularly polarized wave, the rotation of the electric field vector observed in the direction towards the source occurs in a clockwise sense the tip of the vector draws a right-handed helix while propagating in space. In the case of a left circularly polarized wave, the rotation of the electric field vector observed in the direction towards the source is counterclockwise the tip of the propagating vector draws a left-handed helix (Fig. 6.2). Therefore, a circularly polarized wave can be considered as a radiation possessing chirality. 

It is easy to show that a circularly polarized wave can be obtained from the combination of two waves with the same frequency and amplitude, linearly polarized on perpendicular planes and presenting a phase difference of nil. In this manner when the electric field vector of the first wave reaches its maximum, that of the second wave is zero. 

For the sake of the following discussion it is important to observe that, conversely, a linearly polarized wave can be constructed by combining together two waves with the same frequency and amplitude, and exhibiting opposite circular polarization (Fig. 6.3). The phase difference between the two circularly polarized waves determines the orientation of the polarization plane of the resulting linearly polarized wave. 

When the electric field oscillates in amplitude but has a fixed direction, the curve traced out is a straight line and the field remains in one plane. This is called the plane of polarization, and the light is linear or plane polarized. 

When the electric field is of constant amplitude but changes its direction, the point traees out a eirele and the light is called circularly polarized. If both amplitude and direetion change in a regular way, the curve traced out is an ellipse, and the light is elliptically polarized. This is the most general polarized state possible. 

Any state of polarization ean be considered as a combination of two perpendicularly plane polarized waves with different amplitudes and a speeifie phase differenee. Adding two such waves can produce any polarization state. For a fixed result the two waves must be eoherent, so that the phase relation between them remains the same. 

Birefringent materials have a refractive index that depends on the direction of the electric field in the light [5]. They may be single crystals or oriented polymers, either amorphous or 

The optical properties of a birefringent material are shown by a surface called the index ellipsoid or indicatrix. The radial distance from the center to each point on this surface is proportional to the refractive index of light that has its electric displacement D in that radial direction. This is not the refractive index of light that is traveling in that direction. Cross sections of this surface are ellipses. The indicatrix of an isotropic material would be a sphere with circular cross sections. 

Birefringence can be measured directly, by measuring the two refractive indices of the sample and taking the difference, but this is usually inaccurate. Normally the specimen thickness and retardation are measured. Retardation is defined as [( i - nf x (specimen thickness)] in nanometers and is measured using a compensator, which is a crystal plate of known retardation. The specimen to be measured is set to the -45° position between crossed polarizers, and a compensator is inserted in its slot above the specimen but below the analyzer at -h45°. The compensator is adjusted until the specimen is dark, when its retardation is exactly cancelled out by the compensator. If this adjustment is impossible, the sample must be rotated 90° to -t-45°. 

In white light, anisotropic structures may appear brightly colored when viewed in crossed (or parallel) polarizers. These polarization or interference colors depend on the retardation (Section 3.3). The standard sequence of colors, published as the Michel-Levy chart in many texts [6,9,11,18,19] may be used to estimate retardation. 

3-9 o clock and 6-12 o clock, imagining a clock face on the specimen. These directions are referred to as 0° and 90°. 

The electric vector for a plane wave propagating in the positive z-direction can be represented as  

If the right-handed Cartesian frame is chosen (Fig. 2) and is assumed as 

we consider the spatial trace of the electric vector along the z-axis at a certain time, e.g., at t = 0. From Eqs. (4) and (5), it can be seen that the trace of RCP is right-handed as shown in Fig. 2. Moving along the positive z-axis, we find that the 

In order to describe an electro-magnetic wave (light), we need to know the following characteristics  

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A molecule is chiral if it cannot be superimposed on its mirror image (or if it does not possess an alternating axis of symmetry) and would exhibit optical activity, i.e. lead to the rotation of the plane of polarization of polarized light. Lactic acid, which has the structure (2 mirror images) shown exhibits molecular chirality. In this the central carbon atom is said to be chiral but strictly it is the environment which is chiral. 

R.M. Azzam and N.M. Bashara, Ellipsometry and Polarized Light, New York North Holland, 1977. 

The varying actual orientation of molecules adsorbed at an aqueous solution-CCU interface with decreasing A has been followed by resonance Raman spectroscopy using polarized light [130]. The effect of pressure has been studied for fatty alcohols at the water-hexane [131] and water-paraffin oil [132] interfaces. 

At 321 mn there is a vibronic origin marked This has one quantum of v, the antisynnnetric C-H stretching mode, in the upper state. Its intensity is induced by a distortion along This state has B2 vibrational symmetry. The direct product of B2 and A2 is B, so it has B vibronic syimnetry and absorbs x-polarized light. One can also see a 4 6,, vibronic origin which has the same syimnetry and intensity induced by... 

An interesting aspect of two-photon spectroscopy is that some polarization infonnation is obtainable even for randomly oriented molecules in solution by studymg the effect of the relative polarization of die two photons. This is readily done by comparing linearly and circularly polarized light. Transitions to A states will absorb linearly polarized light more strongly than circularly polarized light. The reverse is true of transitions to B ... 

Similar reasoning shows that were one to view along tiieX, Yand Z axes and polarization analyse the signal each time, whether excited by linearly or by naturally polarized light, the total intensity should be given by + 2/. Given eauation (Bl.3.23). if we add its denominator to twice the numerator we find that A... 

Figure B1.3.A.9. Diagram depicting the angles used in scattermg experiments employing linearly and circularly polarized light. The subscripts i and s refer to the incident and scattered beam respectively.

For a circularly polarized light experiment, one can measure the cross sections for either right (r) or left (1) polarized scattered light. Suppose that right polarized light is made incident on a Raman active sample. The general expressions for the Raman cross sections are [176]... 

In analogy with the depolarization ratio for linearly polarized light, the ratio of the two above quantities is known as the reversal coefficient, R(Q, given by... 

Equation (B 1,9.11) is valid only for plane polarized light. For unpolarized incident light, the beam can be resolved into two polarized components at right angles to each other. The scattered intensity can thus be expressed as (figure Bl.9.2)... 

Takmg advantage of the synunetry changes induced by the presence of a surface. Many nonlinear teclmiques rely on the fact that the surface breaks the centrosynuuetrical nature of the bulk. The use of polarized light can also discriminate among dipole moments in different orientations. 

On metals in particular, the dependence of the radiation absorption by surface species on the orientation of the electrical vector can be fiilly exploited by using one of the several polarization techniques developed over the past few decades [27, 28, 29 and 30], The idea behind all those approaches is to acquire the p-to-s polarized light intensity ratio during each single IR interferometer scan since the adsorbate only absorbs the p-polarized component, that spectral ratio provides absorbance infonnation for the surface species exclusively. Polarization-modulation mediods provide the added advantage of being able to discriminate between the signals due to adsorbates and those from gas or liquid molecules. Thanks to this, RAIRS data on species chemisorbed on metals have been successfidly acquired in situ under catalytic conditions [31], and even in electrochemical cells [32]. 

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