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Chapter 7: Problem 9

A stack of optical clements consists of \(N\) layers with index of refraction \(n\) and thickness \(t_{1}\), separated by air gaps \(\left(n_{2}=1\right)\) of thickness \(t_{2}\). A monochromatic plane wave is incident normally, With appropriate thicknesses, a modest number of layers can cause almost total reflection of a given range of frequencies, even for normal \(n\) values \(\left(c, g_{-+}, 1.3

Short Answer

Expert verified
The transfer matrix is \(-\exp(-\lambda \sigma_1)\). Transmitted intensity is \(\operatorname{sech}^2[N \ln(n)]\).

Step by step solution

01

Understand the Problem

We need to find the transfer matrix for a stack of optical layers with specific parameters and then determine the transmission properties when each layer has a quarter-wavelength optical thickness. Two major parts include showing the transfer matrix formula and proving the properties using quarter-wavelength conditions.
02

Define the Transfer Matrix for a Single Layer

For each optical layer, including both refractive and air gaps, the transfer matrix, denoted as \( T \), is given by:\[T = \frac{1}{4n} \left[\begin{aligned}&(n+1)^2 \cos(\alpha_1 + \alpha_2) - (n-1)^2 \cos(\alpha_1 - \alpha_2) \ &+ i \sigma_3 \left[(n+1)^2 \sin(\alpha_1 + \alpha_2) + (n-1)^2 \sin(\alpha_1 - \alpha_2)\right] \ &+ 2 \sigma_1(n^2 - 1) \sin \alpha_1 \sin \alpha_2 \ &- 2 \sigma_2(n^2 - 1) \sin \alpha_1 \cos \alpha_2\end{aligned}\right]\]where \( \alpha_1 = \frac{n \omega t_1}{c} \) and \( \alpha_2 = \frac{b t_2}{c} \).
03

Apply Quarter-Wavelength Condition

Under the quarter-wavelength condition, each optical thickness equals \( \frac{\lambda}{4} \), leading to \( \alpha_1 = \frac{\pi}{2} \) and \( \alpha_2 = \frac{\pi}{2} \). Substitute these into the transfer matrix equations to see simplifications.
04

Simplify the Transfer Matrix

Given \( \alpha_1 = \alpha_2 = \frac{\pi}{2} \), the trigonometric functions simplify:- \( \cos \left(\alpha_1 + \alpha_2\right) = \cos(\pi) = -1 \)- \( \sin \left(\alpha_1 + \alpha_2\right) = \sin(\pi) = 0 \)- \( \cos \left(\alpha_1 - \alpha_2\right) = \cos(0) = 1 \)- \( \sin \left(\alpha_1 - \alpha_2\right) = \sin(0) = 0 \)
05

Construct the Simplified Matrix

Using the simplified trigonometric results:\[T = \frac{1}{4n} \left[- (n^2 - 1) - i\sigma_1 (n^2 - 1) \right]\]This can be further reformulated into exponential form by recognizing the hyperbolic function representation.
06

Derive Exponential Representation

The simplified transfer matrix can be shown to be:\[T = -\exp(-\lambda \sigma_1)\]where \( \lambda = \ln(n) \). This uses the identity of hyperbolic functions.
07

Calculate Transmitted Intensity

For \( N \) layers:\[\frac{|E_{\text{transmitted}}|^2}{|E_{\text{incident}}|^2} = \operatorname{sech}^2[N \ln(n)] = \frac{4n^{2N}}{(n^{2N}+1)^2}\]In the asymptotic limit \( n^{2N} \gg 1 \), the transmission approaches:\[4\exp[-N \ln(n^2)]\]

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Key Concepts

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

Quarter-Wavelength Condition
The quarter-wavelength condition is an essential concept in optics, especially when dealing with layered structures. When each layer, be it a refractive medium or an air gap, has a thickness amounting to a quarter of the wavelength (\( \frac{\lambda}{4} \)) of the incident wave, some fascinating optical properties emerge. This condition optimizes the interference of the light waves reflected within the layers.
When the light reaches the boundary between layers, due to the quarter-wavelength thickness, the reflected waves from different layers combine constructively or destructively, depending on their phase shifts. Here鈥檚 what happens:
  • Constructive interference can lead to increased reflection.
  • Destructive interference minimizes the energy passing through, effectively reducing transmission.
For optical systems using this property, like anti-reflective coatings or high-reflectance mirrors, achieving the quarter-wavelength thickness precisely is key. It ensures the waves add or subtract as desired, which can significantly enhance or suppress reflection.
Reflection and Transmission
Reflection and transmission are fundamental behaviors of light as it interacts with different mediums. When light encounters a boundary between two mediums, part of it is reflected back into the first medium, while another part is transmitted into the second medium. These are essential concepts when discussing the optical transfer matrix. In a stack of multiple layers, like in the given exercise, the light undergoes multiple reflections and transmissions as it moves from one layer to the next. The total effect on the light can be calculated using the transfer matrix approach. This matrix takes into account:
  • The phase shifts due to reflections.
  • The additional path traveled while crossing each layer.
The transfer matrix allows us to predict how much light is transmitted through or reflected from a composite medium. In systems with quarter-wavelength conditions, such matrices simplify considerably, leading to the possibility of near-complete reflection or transmission control.
Refractive Index Layers
Refractive index is a measure of how much light bends, or refracts, when entering a material. Layers with different refractive indices are crucial in manipulating light behavior in optical systems.In the stack described in the exercise, each layer has a refractive index, denoted as \( n \). These varying indices are what cause the light to reflect and transmit differently at each boundary. Here's why it matters:
  • Higher refractive index materials slow down light more, bending it further.
  • The contrast between indices at a boundary determines the reflection and transmission probabilities.
Understanding how refractive index layers interact is crucial to designing optical devices like lenses and anti-reflective coatings. Manipulating these indices across a multi-layer structure allows for precise control over light paths, and in this model, facilitates achieving desired outcomes like reduced transmission or enhanced reflection through strategic interference patterns.

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Most popular questions from this chapter

Two plane semi-infinite slabs of the same uniform, isotropic, nonpermeable, lossless diclectric with index of refraction \(n\) are parallel and separated by an air gap \((n=1)\) of width \(d\). A plane electromagnetic wave of frequency \(\omega\) is incident on the gap from one of the slabs with angle of incidence \(i\). For linear polarization both parallel Io and perpendicular to the plane of incidence, (a) calculate the ratio of power transmitted into the second slab to the incident power and the ratio of reffected to incident power; (b) for \(i\) greater than the critical angle for total internal reflection, sketch the ratio of transmitted power to incident power as a function of \(d\) measured in units of wavelength in the gap.

A plane wave of frequency \(\omega\) is incident normally from vacuum on a semi- infinite slab of matcrial with a complex index of refraction \(n(\omega)\left[n^{2}(\omega)=\epsilon(\omega) / \epsilon_{0} \mid\right.\). (a) Show that the ratio of reflected power to incident power is $$ R=\left|\frac{1-n(\omega)}{1+n(\omega)}\right|^{2} $$ while the ratio of power transmitted into the medium to the incident power is $$ T=\frac{4 \operatorname{Re} n(\omega)}{|1+n(\omega)|^{2}} $$ (b) Evaluate \(\operatorname{Re}\left[i \omega\left(\mathbf{E} \cdot \mathbf{D}^{*}-\mathbf{B} \cdot \mathbf{H}^{*}\right) / 2\right]\) as a function of \((x, y, z)\). Show that this rate of change of encrgy per unit volume accounts for the relative transmitted power \(T_{.}\) (c) For a conductor, with \(n^{2}=1+i\left(\sigma / \omega \epsilon_{0}\right), \sigma\) real, write out the results of parts \(\mathrm{a}\) and \(\mathrm{b}\) in the limit \(\epsilon_{0} \omega<\alpha .\) Express your answer in terms of \(\delta\) as much as possible. Calculate \(\frac{1}{2} \operatorname{Re}\left(\mathbf{J}^{*} \cdot \mathbf{E}\right)\) and compare with the result of part b. Do both enter the complex form of Poynting's theorem?

A stack of optical elements consists of \(N\) layers with index of refraction \(n\) and thickness \(t_{1}\), scparated by air gaps \(\left(n_{2}=1\right)\) of thickness \(t_{2}\). A monochromatic plane wave is incident normally, With appropriate thicknesses, a modest number of layers can cause almost total reflection of a given range of frequencies, even for normal n values (c.g., \(1.3

A monochromatic plane wave of frequency \(\omega\) is incident normally on a stack of layers of various thicknesses \(t_{j}\) and lossless indices of refraction \(n_{,}\) Inside the stack, the wave has both forward and backward moving components. The change in the wave through any interface and also from one side of a layer to the other can be described by means of \(2 \times 2\) transfer matrices. If the electrie field is written as $$ E=E, e^{i k x}+E_{u} e^{-i k x} $$ in each layer, the transfer matrix equation \(E^{*}=T E\) is explicitly $$ \left(\begin{array}{l} E_{+}^{\prime} \\ E^{\prime} \end{array}\right)=\left(\begin{array}{ll} t_{11} & t_{12} \\ b_{21} & t_{22} \end{array}\right)\left(\begin{array}{l} E_{+} \\ E_{-} \end{array}\right) $$ (a) Show that the transfer matrix for propagation inside, but across, a layer of index of refraction \(n_{i}\) and thickness \(t_{j}\) is where \(k_{j}=n_{j} \omega / c, I\) is the unit matrix \(_{+}\) and \(\sigma_{k}\) are the Pauli spin matrices of quantum mechanics. Show that the inverse matrix is \(T^{*}\). (b) Show that the transfer matrix to cross an interface from \(n_{1}\left(xx_{0}\right)\) is $$ T_{\text {intertwoe }}(2,1)=\frac{1}{2}\left(\begin{array}{cc} n+1 & -(n-1) \\ -(n-1) & n+1 \end{array}\right)=I \frac{(n+1)}{2}-\sigma_{1} \frac{(n-1)}{2} $$ where \(n=n_{1} / n_{2}\) (c) Show that for a complete stack, the incident, reflected, and transmitted waves arc related by $$ E_{\text {tram }}=\frac{\operatorname{det}(T)}{t_{22}} E_{\text {inet }} \quad E_{\text {ren }}=-\frac{t_{\text {at }}}{t_{\text {z2 }}} E_{\text {ios }} $$ where \(t_{i j}\) are the elcments of \(T\), the produet of the forward-going transfer matrices, including from the material filling space on the incident side into the first layer and from the last layer into the medium filling the space on the transmitted side.

For each set of Stokes parameters given below deduce the amplitude of the electric field, up to an overall phase, in both linear polarization and circular polarization bases and make an accurate drawing similar to Fig. \(7.4\) showing the lengths of the axes of one of the ellipses and its orientation. (a) \(s_{0}=3, \quad s_{1}=-1, \quad s_{2}=2, \quad 5_{3}=-2\); (b) \(s_{0}=25, \quad s_{1}=0, \quad s_{2}=24, \quad s_{3}=7 .\)
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