Question

You are given four slabs of lossless dielectric, all with the same intrinsic impedance, \(\eta\), known to be different from that of free space. The thickness of each slab is \(\lambda / 4\), where \(\lambda\) is the wavelength as measured in the slab material. The slabs are to be positioned parallel to one another, and the combination lies in the path of a uniform plane wave, normally incident. The slabs are to be arranged such that the airspaces between them are either zero, one-quarter wavelength, or one-half wavelength in thickness. Specify an arrangement of slabs and airspaces such that \((a)\) the wave is totally transmitted through the stack, and \((b)\) the stack presents the highest reflectivity to the incident wave. Several answers may exist.

Short answer

Answer: For total transmission, the arrangement is: 1. Dielectric slab 2. λ/4 airspace 3. Dielectric slab 4. λ/4 airspace 5. Dielectric slab 6. λ/4 airspace 7. Dielectric slab For highest reflectivity, the arrangement is: 1. Dielectric slab 2. λ/2 airspace 3. Dielectric slab 4. λ/2 airspace 5. Dielectric slab 6. λ/2 airspace 7. Dielectric slab

Step-by-step solution

Finding the Transmission Coefficient

To determine the transmission coefficient of an arrangement, we first find the transmission coefficients of the individual dielectric-air interfaces and air-dielectric interfaces. Since the boundary conditions don't change, the transmission coefficients are the same for each interface. Therefore, we can find the overall transmission coefficient as the product of the transmission coefficients of the individual interfaces. We know that for a normally incident wave, the transmission coefficient, \(T\), is given by $$ T = \frac{2\eta_{2}}{\eta_{1} + \eta_{2}}, $$ where \(\eta_{1}\) and \(\eta_{2}\) are the intrinsic impedances of the first and second media, respectively. For total transmission, we want the overall transmission coefficient to be equal to 1. A set of four slabs and airspaces should be connected to allow this to happen.

Arrangement for Total Transmission

To achieve total transmission through the stack, we'll arrange the four dielectric slabs with quarter-wavelength airspaces between them: 1. Dielectric slab 2. \(\lambda/4\) airspace 3. Dielectric slab 4. \(\lambda/4\) airspace 5. Dielectric slab 6. \(\lambda/4\) airspace 7. Dielectric slab This arrangement helps to cancel out the reflections that occur at each interface, such that the transmitted wave builds up constructively, and thus all the wave energy passes through the stack.

Arrangement for Highest Reflectivity

Now, we are interested in an arrangement that presents the highest reflectivity for the incident wave. In this case, we want the magnitude of the overall reflection coefficient to be maximum. For the highest reflectivity, we'll arrange the dielectric slabs with half-wavelength airspaces between them: 1. Dielectric slab 2. \(\lambda/2\) airspace 3. Dielectric slab 4. \(\lambda/2\) airspace 5. Dielectric slab 6. \(\lambda/2\) airspace 7. Dielectric slab This arrangement leads to reflections that add up constructively and result in the maximum reflectivity of the stack.

Key concepts

Intrinsic Impedance

Intrinsic impedance, often represented by the symbol \(\eta\), is a fundamental property of a medium that defines how electromagnetic waves propagate through it. It tells us how the electric and magnetic fields of the wave relate to each other.
This intrinsic impedance can vary depending on the material properties of the medium, such as permittivity and permeability.
In free space, the intrinsic impedance is approximately 377 ohms, but in dielectric materials, it can be different.When waves encounter a boundary between two different media, the intrinsic impedance determines the amount of wave energy that gets reflected and the amount that gets transmitted.
Therefore, knowing the intrinsic impedance is crucial for designing systems that manipulate waves, such as those involving dielectric slabs. In such systems, matching the intrinsic impedance between different layers can minimize reflections and optimize transmission.

Wavelength

Wavelength, denoted by \(\lambda\), refers to the distance over which the wave's shape repeats.
In the context of dielectric materials, the wavelength is different from that in free space due to the material's refractive index.
The properties of the dielectric influence how the electromagnetic waves behave, altering both their speed and wavelength as compared to a vacuum.When waves enter a dielectric slab, the wavelength becomes \(\lambda/n_m\), where \(n_m\) is the refractive index of the material.
This change in wavelength impacts how we design systems involving stacks of dielectric materials.
For achieving specific constructive or destructive interference effects, it’s essential to adjust the thickness of the slabs, often using ratios like a quarter or half wavelength, to control how the wave fronts interact.

Dielectric Slabs

Dielectric slabs are materials with insulating properties that don't conduct electricity but can store electrostatic energy.
They are used in a variety of applications to control and manipulate electromagnetic waves.
The intrinsic impedance and thickness of these slabs are vital in determining how waves will behave as they travel through these materials.In the given exercise, using slabs with a thickness of \(\lambda/4\), enables specific control over wave interference patterns.
This ensures either maximized transmission by constructive interference or heightened reflectivity by destructive interference.
By strategically arranging dielectric slabs and airspaces, one can engineer systems to craft desired wave behaviors—such as minimal reflection or perfect transmission through a stack.

Transmission Coefficient

The transmission coefficient, symbolized as \(T\), quantifies how much of an incident wave's energy passes through a medium or interface.
It ranges from 0 to 1, with 1 indicating full transmission without any energy lost to reflection.To calculate the transmission coefficient, you can use the equation: \[ T = \frac{2\eta_{2}}{\eta_{1} + \eta_{2}} \]where \(\eta_{1}\) and \(\eta_{2}\) are the intrinsic impedances of adjacent materials.
In scenarios where transmission through multiple slabs and interfaces is needed, the overall transmission coefficient is determined by multiplying the transmission coefficients of all individual interfaces.For complete transmission—as in the example's first arrangement, where \(T = 1\)—the design leverages specific thicknesses and spacing of dielectric and air slabs, thereby ensuring wave propagation occurs without internal reflections disrupting the passage of energy.