Plane Wave Incidence on an Interface: Program WAVES

C.W. Trueman

September 12, 2000

Updated October 10, 2000

Program WAVES animates plane wave incidence on a dielectric interface. The program is very easy to use. These notes describe how to use the program and outline a series of wave demonstrates that can be done with WAVES. Students should download the program and run the animation associated with each figure in these notes.

You can download the program by clicking this button: Get WAVES. You can download CPLOT.EXE by clicking this button: Get CPLOT. You can download RPLOT.EXE by clicking this button: Get RPLOT.

Make sure you have the latest copy of CPLOT and RPLOT as these programs were updated on Oct. 10, 2000.

Program WAVES graphs the magnitude of the electric field as a function of position (x,y) for a plane wave incident on an interface between two dielectrics. In medium #1, the total field is graphed, consisting of the incident field plus the reflected field. In medium #2 the transmitted field is graphed. The program uses the reflection and transmission coefficients given in Reference [1]. The geometry of the problem is as follows. A wave travels in the xy plane in medium #1. The boundary is a yz plane, with unit normal vector in the x direction. On the far side of the boundary we have medium #2. The wave is partially reflected back into medium #1, so the field there is the sum of the incident wave plus the reflected wave. The wave is partially transmitted into medium #2, so the field there is equal to the transmitted wave. WAVES draws a color contour map of the magnitude of the electric field in the xy plane, showing the field on both sides of the boundary between the two materials. As time advances, the waves move and program WAVES animates their movement.

WAVES creates an output file that contains the amplitude of the field as a function of position xy. Note that the amplitude is the magnitude of the phasor and is independent of time. When you exit from the WAVES program with F1, WAVES writes file efield.tbl, and starts cplot, which displays the amplitude as a function of position. This is handy for graphing the standing wave, as discussed under “Graphing the Standing Wave Pattern”, below.

Fig. 1 The WAVES menu.

The WAVES Menu

Start WAVES by double-clicking on the program name in a directory window. Then the program shows the menu of Fig. 1. There are various boxes in which you can enter the parameters of your problem.

The first data box asks for the frequency in MHz. Choose the angle of incidence using the box at the top. If the angle is zero degrees, the wave travels parallel to the x-axis, which is horizontally across the screen. Then the wave is normally incident on the boundary between the two media. If the angle is 90 degrees, the wave travels parallel to the y-axis, which is vertically up the screen. Then the wave is traveling parallel to the interface between the two media.

Specify either “perpendicular” or “parallel” polarization as defined in Ref. [1].

Choose the relative permittivity of medium #1, which is the material to the left of the interface. Medium #1 is lossless, that is, has zero conductivity. Choose a relative permittivity and conductivity for medium #2.

The last two boxes permit you to define the extent of the region in the xy plane that WAVES will use in graphing the field. This is done in terms of free-space wavelengths. The interface is always at x=0. The region to the left of the interface is medium #1. Enter the x-coordinate of the left edge of the screen, which must be negative. In Fig. 1, the user is asking to see two (free-space) wavelengths of distance in medium #1. Similarly, enter the size of region #2 in (free-space) wavelengths, chosen as one in Fig. 1. Then the width of the screen is used to display three free-space wavelengths of distance, with the interface between the two media two wavelengths from the left edge. We will see three complete cycles of the plane wave, if both media are free-space.

The function keys are used to start the simulation, and to exit. F1 exits the program. Typing the F10 key starts the simulation. An information screen shows the angle of incidence and the angle of transmission into medium #2, from Snell’s Law. Also, the Brewster angle is shown. For the parallel polarization, if the incidence angle equals the Brewster angle there will be total transmission into medium #2. Finally, the critical angle is shown if there is one. If the relative permittivity of medium #1 is greater than that of the (lossless) medium #2, then for incidence at an angle greater than the critical angle, the wave will be completely reflected back into medium #1.

Typing “return” advances to the next information screen. This gives the value of the intrinsic impedance of both materials, and the complex value of the reflection coefficient and of the transmission coefficient. This is useful for checking homework answers!

Typing “return” starts the simulation. Type “s” during the simulation to stop. Use can use the Edit-Copy function at the upper left of the screen to copy the screen as a bit map, and paste it into a document such as this one. Type any key (which is the “any key”?) to return to the menu.

Fig. 2 A plane wave traveling from left to right across the screen.

The WAVES Display

Fig. 2 is a simple display made by program WAVES. The screen shows a region of the xy plane with the x axis running horizontally along the screen and the y axis vertically up the screen. There is an interface between two media at x=0, shown by a black line. Medium #1 is to the left of the interface and is a lossless dielectric with relative permittivity . Medium #2 is located to the right of the interface and has relative permittivity and conductivity mS/m. A plane wave is incident on the interface from the left, traveling at an angle of to the normal to the interface. Thus =0 corresponds to normal incidence on the interface, as in Fig. 2. To obtain Fig. 2 set the frequency to 300 MHz and the angle of incidence to zero degrees. Set the relative permittivity of medium #2 to unity and the conductivity to zero. Program WAVES calculates the reflection and transmission coefficients, then evaluates the field in each region and plots it as a color contour map at each instant of time. As time advances the waves appear to move across the screen. The following describes various wave phenomena that can be demonstrated with WAVES.

A “Traveling Wave”

We can demonstrate a simple “traveling wave” by making both media the same, and setting the angle of incidence to zero. Set the frequency to 300 MHz. Choose an incidence angle of 0 degrees, and set the relative permittivity of both media to 1, and the conductivity to zero. This fills the whole screen with free space. Run the simulation and see a plane wave traveling from left to right. Fig. 2 shows the plane wave at one instant of time. The value of the electric field is shown on a color scale with “blue” representing –20 V/m, cyan –10 V/m, green 0V/m, yellow 10 V/m and red 20 V/m. The “incident” wave in WAVES has amplitude 10 V/m by definition. We see a wave that has peaks of + 10 volts (yellow) and troughs of –10 volts (cyan). The wavelength is the distance between two yellow peaks, or between two blue troughs. As time advances the peaks and troughs advance from left to right across the screen, and so the wave “travels”. The vertical black line represents the interface between the two media. Since both media are free space, the wave simply flows through the (imaginary) interface.

Traveling Wave in a Dielectric

Change the relative permittivity of both media to 3 and re-run the simulation to obtain Fig. 3.

Fig. 3 The plane wave in a dielectric of relative permittivity 3 has a shorter wavelength.

Compare the wave in Fig. 3 with that in Fig. 2. It is clear that the wavelength in the dielectric region is shorter than in free space. Also, if you watch the animation carefully, the wave in the dielectric travels more slowly than the wave in free space.

Total Reflection and Standing Waves

The waves in Fig. 2 and Fig. 3 are “traveling waves” and are the simplest kind of wave. When medium #1 and medium #2 have different permittivities then there is a reflected wave from the boundary. Then in medium #1, the field is more complex, being the superposition of a wave traveling to the right plus a wave traveling got the left.

The easiest way to obtain almost complete reflection at the boundary between the two materials is to make material #2 a very good conductor. Set the permittivity of material #2 to unity and the conductivity to, say, 1,000,000 mS/m. The reflection coefficient is then –0.9998, close to the value of –1 for a perfect conductor. Fig. 4 shows the resulting field.

Fig. 4 A standing wave forms when material #2 is a good conductor.

Fig. 4 is quite different from Fig. 2 or Fig. 3. Also, in the animated display, the peaks and troughs in Fig. 4 do not move across the screen as in Fig. 2 or 3. They oscillate in place and the wave is called a “standing wave”.

Fig.4 shows the field in medium #1 at an instant of time where there are large maxima(red) and minima(blue). The field in medium #2, the good conductor, is of course equal to zero. As time progresses the wave does not travel across the screen. Instead the green bars of zero field remain in place and the peaks and troughs oscillate up and down without moving. This is a standing wave. A pure standing wave is composed of equal-amplitude waves traveling in opposite directions. The incident wave travels to the right and the reflected wave travels to the left.

Fig. 5 A standing wave in medium #1 when medium #2 is a lossless material with a high relative permittivity of 1000.

Another way to obtain a standing wave is keep medium #2 lossless with zero conductivity, and to make the relative permittivity of region #2 large, say, equal to 1000. This leads to a large impedance mismatch between the two regions, so little field is transmitted into the second medium. The reflection coefficient is –0.94, and the transmission coefficient 0.06. The wave is almost fully reflected back into medium #1. Fig. 5 shows that the standing wave in this case is very similar to that for the perfect conductor in Fig. 4.

Graphing the Standing Wave Pattern

When you exit from WAVES with F1, the program writes a file called efield.tbl, for input to CPLOT, and then starts CPLOT. This file contains the amplitude of the field as a function of position. Note that the amplitude of the field is independent of the time. It is the magnitude of the total field phasor in medium #1, incident plus reflected, and the magnitude of the transmitted field phasor in medium #2. Fig. 6 shows the amplitude when medium #1 is free space and medium #2 is a good conductor with =1 and =1,000,000 mS/m, as in Fig. 4.

Fig. 6 The amplitude of the field graphed with CPLOT using efield.tbl.

Fig. 6 shows the amplitude of the field as graphed by CPLOT. You can “read back” values from the display by clicking the mouse on the color map. The values appear in the lower left corner. This is handy for getting the approximate location and field strength of the minima and maxima in the standing wave pattern. The values are approximate because CPLOT finds the point nearest the mouse in the input data, and the input data does not usually contain precisely the maxima or minima.

Fig.7 The standing-wave pattern graphed with RPLOT from CPLOT’s “cut menu” output.

You can use the F8 “cut menu” in CPLOT to graph the standing wave pattern as a function of distance x. Type F8 and then type F5 to change the cut to constant-y. Then type F3. CPLOT writes data file cplot.rpl then starts RPLOT to graph it, to obtain Fig. 7. In RPLOT you can click the mouse on the curve to “read back” values, as well. You can get the approximate locations of the minima this way. Note in Fig. 7 that the minima are somewhat squared off and imperfect. RPLOT joins the points in the input data file with straight lines; the input data does not sample the field finely enough in the minima.

Partial Reflection and Partial Transmission

Make the relative permittivity of region #1 equal to one and that of region #2 equal to 3. Then we have partial reflection. The reflection coefficient is –0.27 and the transmission coefficient is 0.73. Most of the wave is transmitted into medium #2.

Fig. 8 Partial reflection, with a reflection coefficient of –0.27.

Fig. 8 shows the field. In medium #2 we see the field traveling away to the right with the shorter wavelength characteristic of a wave in a dielectric. In medium #1, the wave looks very much like a traveling wave, but if you watch the animation closely the phase fronts do not progress uniformly across the screen. This behavior is enhanced if the relative permittivity of medium #2 is increased.

Fig. 9 Partial reflection with a reflection coefficient of –0.5.

Fig. 9 shows the field at one instance of time with a relative permittivity for medium #2 of 9, hence a reflection coefficient of –0.5. The field to the left of the boundary has troughs of –15 V/m, which shows up as a blue region, and peaks of 15 V/m, graphed as orange. As time progresses the peak of the wave does not progress uniformly across the screen. If the relative permittivity of medium #2 is further increased, the behavior becomes more and more like the standing wave of Fig. 4.

Transmission into a Lossy Dielectric

If medium #2 is a dielectric with some conductivity it is called a “lossy dielectric”. It absorbs energy. Then the wave transmitted into medium #2 decreases in amplitude as it travels into medium #2. We can demonstrate this by setting the relative permittivity of medium #1 to 1 for free space, and making medium #2 lossy with relative permittivity 1 and conductivity, say, 3 mS/m. With this small conductivity the reflection coefficient magnitude is only 0.004, and the transmission coefficient 0.996, so most of the wave is transmitted into the second medium. Set the right-hand boundary of the space to 2 wavelengths to see farther into medium #2 and run the simulation.

Fig.10 Transmission from a lossless medium at left into a lossy medium at right.

Fig. 10 shows the wave progressing from free space at left into the lossy region at right. In the region at left the amplitude of the wave does not change as the wave travels from left to right across the screen. The two yellow bars at left are of the same intensity, representing a field strength of about 10 V/m. But after the wave crosses the black line representing the boundary, the amplitude of the wave rapidly diminishes to zero. Thus the yellow maximum just to the right of the interface in Fig.10 is almost as large as the yellow maxima to the left. But moving a wavelength deeper into the 2^nd material, the yellow maximum at right is almost invisible and the wave has been absorbed by the material.

The material in to the right of the boundary in Fig. 10 behaves like microwave absorber material. Its permittivity is the same as free space, and its small conductivity absorbs the microwave energy. The reflection coefficient is only 0.004 so little of the wave is reflected back into the free space region.

Fig. 11 A plane wave in air incident on a material with the parameters of real ground.

Dielectrics generally have relative permittivities of 3 or more. Fig. 11 shows transmission from free space into a dielectric having the electrical parameters of real ground, namely relative permittivity 15 and conductivity 10 mS/m. Because there is a large impedance mismatch at the surface, much of the wave is reflected back into the air. The reflection coefficient is about 0.6. We see a wave in medium #1 which looks very much like a standing wave. With a transmission coefficient of 0.4, the wave transmitted into material #2 has only 10 V/m amplitude at the surface, and it decreases in amplitude as we progress into medium #2. Thus the yellow bars in material #2 become fainter as we get further from the surface.

Waves Traveling Obliquely

If the incidence angle is changed to 30 degrees, then the plane wave travels diagonally up the screen from lower left to upper right, as shown in Fig. 12. Change the incidence angle to see the wave travel in various directions.

Fig. 12 A plane wave in free space traveling at 30 degrees to the normal.

In Fig. 12 both of the materials are free space. The plane wave travels at an incidence angle of 30 degrees to the unit normal to the surface. The wave looks very much like that of Fig. 2, except that it travels in a different direction.

Oblique Incidence – Parallel Polarization

Keep the relative permittivity of region #1 equal to unit, but change the relative permittivity of region 2 to 3, and set the angle of incidence to 30 degrees. Use perpendicular polarization, which makes the electric field vector perpendicular to the computer screen. Thus the field is parallel to the planar interface between the two media. The reflection coefficient is –0.31.

Fig13 Incidence at 30 degrees for the perpendicular polarization.

Fig. 13 shows the field. The plane wave travels upwards to the right in medium 1. The wavefronts are not straight lines, however, because there is a reflected wave from the interface. The pattern has pink regions of high field(hard to see!) in the maxima(yellow), and blue regions of low field in the minima(cyan), reminiscent of a standing wave pattern. Note that as we move from one side of the interface to the other, the colors are continuous. Thus the field satisfies the boundary condition that the tangential electric field must be continuous across the interface. In medium #2 a plane wave moves away from the interface, propagating at a different angle, according to Snell’s Law. The wavelength is shorter in the dielectric, as expected.

Fig. 14 Incidence at 60 degrees for perpendicular polarization.

Change the angle of incidence to 60 degrees. The reflection coefficient is now -0.5, and the standing wave pattern in medium #1 is much more pronounced, as in Fig. 14. As we increase the angle of incidence the reflection coefficient rises and the standing wave pattern becomes more and more clear.

Fig. 15 The standing wave pattern for incidence at 60 degrees on a medium with relative permittivity 9, in the perpendicular polarization.

As we increase the relative permittivity of medium #2, the standing wave pattern becomes more pronounced. Thus with the relative permittivity of medium #2 set at 9, and an angle of incidence of 60 degrees, we have the pattern of Fig. 15. There are clear troughs(blue) and peaks(red). As time progresses, the pattern moves upward on the screen, moving parallel to the interface between the two materials. This behavior is a standing wave in the direction perpendicular to the interface(x) and a traveling wave along the interface(y). We encounter similar behavior in waveguides[1].

Parallel Polarization

Reset the relative permittivity of material #2 to 3, and the incidence angle to 30 degrees. Set the polarization to parallel, meaning that the electric field vector lies in the plane of the computer screen. Note that since the electric field is also parallel to the wave front of each wave, the field is not parallel to the interface between the two materials. Program WAVES graphs the value of the electric field vector instant by instant as time advances. But since the electric field is not parallel to the interface in the parallel polarization, we do not expect it to be continuous across the interface.

Fig. 16 Incidence at 30 degrees for the parallel polarization.

Fig. 16 shows the field in the parallel polarization, where the reflection coefficient from the interface is –0.22. Compare this with the corresponding picture for perpendicular polarization, Fig. 13. The reflection coefficient is smaller and the standing-wave pattern is less pronounced. As we increase the angle of incidence the reflection coefficient decreases. At 60 degrees incidence angle, the reflection coefficient is zero and there is total transmission into medium #2. This angle is called the “Brewster angle”.

Fig. 17 Incidence at the Brewster angle, parallel polarization.

Fig. 17 shows the field when the angle of incidence equals the Brewster Angle. Note that there is no standing wave in medium #1. The wave is a pure traveling wave with straight wavefronts. In medium #2 the wave travels away from the interface with a smaller amplitude than in medium #1, because the transmission coefficient is 0.58. If we have total transmission, why is the transmission coefficient not equal to unity? The answer is that the power density is conserved, so the power traveling towards the interface is equal to the power traveling away from the interface. In the higher-permittivity medium #2, a lower-amplitude field carries the same power.

Note in Fig. 17 that the colors representing the field are not continuous across the interface. This is because the program graphs the instantaneous value of the electric field, irrespective of its vector direction. In the parallel polarization the field vectors are parallel to the plane of the screen, not parallel to the interface between the two media, and so the field is not continuous across the boundary.

Change the polarization to perpendicular, and look at the field with the angle of incidence equal to the Brewster angle, shown in Fig. 14. The reflection coefficient is –0.5 and there is a pronounced standing wave in medium #1.

Transmission from a Dielectric into Air

Consider a wave in medium #1 with relative permittivity three, traveling into medium #2 which is free space with relative permittivity unity. Fig. 16 shows the field for normal incidence. The reflection coefficient is +0.27.

Fig. 18 Transmission from a dielectric into air.

In Fig. 18, we see that the wavelength in the dielectric is shorter than that in air, as expected. Since the reflection coefficient is small, the wave in the dielectric is largely a traveling wave. The transmission coefficient is about 1.3, so the wave transmitted into the air has a larger amplitude than the wave incident on the interface in the dielectric.

For an angle of incidence of 20 degrees, the reflection coefficient decreases to 0.20; for 30 degrees rises to 0.5; for 33 degrees to 0.63; and for 35 degrees to 0.85. The “critical angle” is 35.26 degrees, at which the reflection coefficient is unity. For angles of incidence larger than the critical angle we have total reflection back into medium #1.

Fig. 19 The field for a wave incident at 35.25 degrees, very close to the critical angle.

Set the incidence angle to 35.25 degrees, just less than the critical angle. The reflection coefficient is 0.96. The field is that in Fig. 19. In medium #1 we have a standing wave pattern in the direction perpendicular to the interface. As time progresses the pattern travels up the screen parallel to the interface. In medium #2 we have a wave traveling almost parallel to the interface, traveling away at 88.47 degrees to the normal.

Fig. 20 The angle of incidence is 35.27 degrees, just larger than the critical angle.

If we change the angle of incidence to 35.27 degrees, a little larger than the critical angle, then the wave is shown in Fig. 20, which is very similar to Fig. 19. There are some subtle differences, however. The reflection coefficient magnitude is now exactly equal to unity and the wave is fully reflected. In medium #2 the wave now travels parallel to the y axis, vertically up the screen. In the x direction the wave attenuates exponentially with distance, but so slowly that we do not see much change in Fig. 18 in the field strength as we go farther from the boundary into medium #2.

Fig. 21 The field with incidence angle 40 degrees.

When the angle of incidence is 40 degrees, the field is that shown in Fig. 21. Now the field decays rapidly as we go into medium #2, and is negligible in value at the right hand side of the screen. The wave in medium #2 is called a surface wave. It travels along the surface, up the screen, and decays rapidly with distance from the surface. Surface waves are frequently encountered in electromagnetics. Fiber optic cables carry surface waves at interfaces where the permittivity changes. At HF frequencies(2-30 MHz), the surface of the earth carries a surface wave that can propagate the field over the horizon.

Conclusion

Program WAVES is very easy to use. These notes have described a variety of experiments that can be done with WAVES. Each figure in these notes is one frame in the animation of the wave behavior as a function of time. With a little experience in interpreting the animated color contour map of the field strength as time advances, many useful insights into wave interactions at dielectric surfaces can be gained. Students are encouraged to download the program and look at the animation.

Reference

[1] B.S. Singh Guru and H.R. Hiziroglu, “Electromagnetic Field Theory Fundamentals”, PWS Publishing Company, Boston, 1998.