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Chapter 6: Problem 7

Maximum Retardation in Quartz. Quartz is a positive uniaxial crystal with \(n_{e}=1.5\) and \(n_{0}=1.544\). (a) Determine the retardation per mm at \(\lambda_{0}=633 \mathrm{~nm}\) when the crystal oriented such that retardation is maximized. (b) At what thickness(es) does the crystal act a quarter-wave retarder?

Short Answer

Expert verified
(a) The maximum retardation per mm is 69.5 waves/mm. (b) The thickness for a quarter-wave retarder is approximately 3.6 渭m for the first instance.

Step by step solution

01

Understand Retardation

Retardation (or optical path difference) in a uniaxial crystal arises from the difference in the refractive indices for the extraordinary and ordinary rays, given by the formula \( R = (n_o - n_e)\cdot t \), where \( t \) is the thickness of the crystal. For maximum retardation, let's substitute the known values.
02

Calculate Maximum Retardation per mm

We are given \( n_e = 1.5 \) and \( n_o = 1.544 \). The maximum retardation per mm will be when the crystal thickness \( t \) is 1 mm. Calculate this using:\[R_{max} = (n_o - n_e) \times 1 \, \text{mm} = (1.544 - 1.5) \times 1 \, \text{mm} = 0.044 \, \text{mm}\]Since the problem asks for the result in terms of wavelength \( \lambda_0 \), we must convert mm to nm by multiplying \( 0.044 \, \text{mm} \times 10^6 \, \text{nm/mm} = 44,000 \, \text{nm} \). However, we want it per mm, hence:\[R_{max} = \frac{44,000 \, \text{nm}}{633 \, \text{nm}/\text{wave}} = 69.5 \, \text{waves/mm}\]
03

Quarter-Wave Retarder Condition

A quarter-wave retarder causes a retardation of \( \frac{\lambda_0}{4} \). Solve for the thickness \( t \) where the optical path difference \( R = \frac{633}{4} \). Using the relation:\[R = (n_o - n_e)\cdot t = \frac{633}{4} \, \text{nm}\]Solve for \( t \):\[t = \frac{633}{4 \times (n_o - n_e)} = \frac{633}{4 \times 0.044} = 3596.59 \, \text{nm} \approx 3.6 \, \text{渭m}\]
04

Considering Higher Orders

Since quarter-wave retardation can also occur for delays \( \frac{(2m+1)\lambda_0}{4} \) where \( m \) is an integer, the thickness can be generalized as:\[t_m = \frac{(2m+1)\cdot 633}{4 \times 0.044} \, \text{nm}\]For each possible \( m \), this will yield different thicknesses.

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Key Concepts

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

Uniaxial Crystal
A uniaxial crystal, like quartz, has a unique optical property in which it possesses two principal refractive indices. These indices relate to the ordinary and extraordinary rays within the crystal. In a uniaxial crystal, light traveling in specific directions experiences different velocities depending on the polarization of the light. This means:
  • The ordinary ray travels at a speed dictated by the ordinary refractive index, typically denoted as \(n_o\).
  • The extraordinary ray follows a path influenced by the extraordinary refractive index, represented as \(n_e\).
Quartz, being a positive uniaxial crystal, has a situation where the ordinary refractive index \(n_o\) is greater than the extraordinary refractive index \(n_e\). This characteristic impacts how light is refracted and leads to the creation of optical phenomena such as retardation.
Refractive Index
The refractive index is a measure that describes how light, or any other radiation, propagates through a medium. It is a fundamental concept in optics and helps determine how much a light wave is bent or refracted when entering a material. For a material, the refractive index is defined by:
  • \(n = \frac{c}{v}\)
where \(c\) is the speed of light in a vacuum, and \(v\) is the speed of light in the material. In uniaxial crystals, the difference between the refractive indices of the ordinary and extraordinary rays (\(n_o\) and \(n_e\)) gives rise to optical path differences. These differences cause effects such as optical retardation, essential in designing optical elements like wave plates. Thus, knowing the refractive indices of a material is crucial for predicting how it will interact with light.
Quarter-Wave Plate
A quarter-wave plate is an optical device designed to create a phase shift of 90 degrees (one-quarter wavelength) between the orthogonal polarizations of a light wave passing through it. This phase shift converts linearly polarized light into circular or elliptical polarized light, depending on orientation. The operation principle of a quarter-wave plate is based on:
  • The optical path difference set to \(\frac{\lambda}{4}\), where \(\lambda\) is the wavelength of light used.
  • The thickness of the quartz, which must be carefully chosen to achieve the desired phase shift.
By exploiting the differences in refractive indices within uniaxial crystals, specific thicknesses allow for this precise retardation, making quarter-wave plates vital in multiple applications including microscopy, telecommunications, and the study of polarizations.
Optical Path Difference
The optical path difference (OPD) is a vital concept in understanding how light behaves as it travels through a crystalline medium. It's the difference in the path length traveled by two light beams inside a material with varying refractive indices. Mathematically, OPD is expressed as:
  • \(R = (n_o - n_e) \cdot t\)
where \(R\) is the retardation, \(n_o\) is the ordinary refractive index, \(n_e\) is the extraordinary refractive index, and \(t\) is the thickness of the material. The OPD is crucial because it determines the phenomenon of interference, which can result in striking patterns or changes in light polarization. In applications, controlling the OPD allows engineers and scientists to design sophisticated optical systems, including those that use quarter-wave plates for polarization control.

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Most popular questions from this chapter

Half-Wave Retarder. Consider linearly polarized light passed through a half- wave retarder. If the polarization plane makes an angle \(\theta\) with the fast axis of the retarder, show that the transmitted light is linearly polarized at an angle \(-\theta\), i.e., it is rotated by an angle \(2 \theta\). Why is the half-wave retarder not equivalent to a polarization rotator?

Double Refraction. An unpolarized plane wave is incident from free space onto a quartz crystal \(\left(n_{c}=1.553\right.\) and \(\left.\pi_{0}=1.541\right)\) at an angle of incidence \(30^{\circ}\). The optic axis lies in the plane of incidence and is perpendicular to the direction of the incident wave before it enters the crystal. Determine the directions of the wavevectors and the rays of the two refracted components.

Polarization Rotation by a Sequence of Linear Polarizers. A wave that is linearly polarized in the \(x\) direction is transmitted through a sequence of \(N\) linear polarizers whose transmission axes are inclined by angles \(m \theta(m=1,2, \ldots, N ; \theta=\pi / 2 N)\) with respect to the \(x\) axis. Show that the transmitted light is linearly polarized in the \(y\) direction but its amplitude is reduced by the factor \(\cos ^{N} \theta\). What happens in the limit \(N \rightarrow \infty\) ? Hint: Use Jones matrices and note that $$ \mathbf{R}[(m+1) \theta] \mathbf{R}(-m \theta)=\mathbf{R}(\theta)_{1} $$ where \(\mathbf{R}(\theta)\) is the coordinate transformation matrix.

Wave Retarders in Tandem. Write the Jones matrices for: (a) A \(\pi / 2\) wave retarder with the fast axis along the \(x\) direction. (b) A \(\pi\) wave retarder with the fast axis at \(45^{\circ}\) to the \(x\) direction. (c) A \(\pi / 2\) wave retarder with the fast axis along the \(y\) direction. If these three retardens are placed in tandem, with (c) following (b) following (a), show th the resulting device introduces a \(90^{\circ}\) rotation. What happens if the order of the three retarde is reversed?

Transmission Through a LiNbO \(_{3}\) Plate. Examine the transmission of an unpolarized HeNe laser beam \(\left(\lambda_{0}=633 \mathrm{~nm}\right)\) normally incident on a LiNbO \(_{3}\) plate \(\left(n_{e}=2.29, n_{0}=2.20\right)\) of thickness \(1 \mathrm{~cm}\), cut such that its optic axis makes an angle \(45^{\circ}\) with the normal to the plate. Determine the lateral shift at the output of the plate and the retardation between the ordinary and extraordinary beams.
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