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Plane Wave – Normal incidence (Air-Dielectric-Air)


Plane Wave - Reflection Coefficient for Normal Incidence

The Physics

Whenever a plane wave propagating along a region 1 with intrinsic impedance η1 encounters a given region 2 with intrinsic impedance η2 , a scattering problem has to be solved in order to determine the scattering coefficients, in order to determine the behavior of the fields at the interface of both media. With such information it is possible to determine how much of the fields is reflected to region 1 and how much goes through region 2.

Plane wave normally illuminating a dielectric slab
Figure 2: Transmitted and reflected fields for a plane wave travelling in free space and illuminating a dielectric slab.

The general approach to retrieve the scattering parameters consists of mode matching techniques, which in general can be obtained by imposing the continuity of the tangential components of the fields at the interface of both media.

The Model

This is a very simple model in which a plane wave (generated using waveguide ports) with linear polarization travels through a volume of air divided by a dielectric slab, as shown in Fig. 3. The permittivity of the dielectric is parameterized to allow the reflection coefficient to be calculated for a range of different scenarios. The model is simulated with the frequency domain solver at a frequency of 1 GHz.

Air-Dielectric-Air model for normal plane wave incidence
Figure 3: Air-Dielectric-Air model for plane wave incidence. Waveguide ports (red) are used to mimic plane waves. The dielectric slab is dark blue.

 

Model Construction Watch Video

Download Model File (Student Edition) Download Model File

Discussion of Results

In Fig. 4 we show – for a fixed frequency – the dependence of the reflection coefficient (linear) on the permittivity of the dielectric substrate with thickness

Reflection coefficient as a function of the permittivity of the slab
Figure 4. Reflection coefficient as a function of the permittivity of the slab for a fixed frequency.

 

The solid line represents the analytical solution obtained from solving the scattering problem and the discrete symbols are obtained using CST STUDIO SUITE Student Edition. As can be seen, there is excellent agreement between CST STUDIO SUITE and the analytical formula. It can be seen that the reflection hits a null several times at resonances, and the obtained curve has a “growing-sinusoidal” behavior i.e. as permittivity increases the maximum value of each sinusoid gets closer to 1, which means the reflection peaks also get bigger. This happens because as permittivity increases, the mismatch between the intrinsic impedance of free-space and the dielectric gets bigger, therefore the adaptation is worse and we have greater reflection.

Fig. 5 shows a carpet plot of the real part of the y-component of the magnetic field along the whole system. This carpet plot is obtained for a case in which the permittivity of the slab is 4. This corresponds to one of the nulls in the reflection coefficient, and so reflection is zero and the amplitude is the same on either side of the slab.

y-component of the Magnetic Field for a slab permittivity of eps=4
Figure 5: Carpet Plot of the y-component of the Magnetic Field when the permittivity of the slab is 4.

Fig. 6 shows the carpet plot is shown for the case at which the permittivity is 10. As seen in Fig. 4, the reflection for this permittivity value is nearly 0.78, so the transmission is clearly affected.

y-component of the Magnetic Field for a slab permittivity of eps=10
Figure 6: Carpet Plot of the y-component of the Magnetic Field when the permittivity of the slab is 10.

Questions

  • Which symmetry plane conditions (i.e. where the field vectors are normal to the symmetry plane) can be used to speed up the simulation?
  • How could you measure the wavelength λd inside the dielectric in order to prove that it is such that λd=λ0ε? Using the curve tools, draw a line along the Z axis inside the dielectric slab. From “Template Based Post Processing” (2D and 3D Field Results), use the “Field on Curve” option to plot the real part of the y-component of the magnetic field.

References

  • [1] D.M. Pozar, Microwave Engineering, 4th Edition, John Wiley & Sons, pp. 28-30, 45-46
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