# Ibn al-Haytham to Maxwell: A Long Road

As the Chinese proverb says “The journey of a thousand miles begins with a single step”. The journey that started with Ibn al-Haytham experimenting with his Camera Obscura in the eleventh century was completed eight hundred years later by James Clerk Maxwell and Heinrich Hertz. While Maxwell laid down the mathematical framework that described the behavior of Electromagnetic waves, Hertz conclusively proved the existing of these invisible waves through his experiments. There were several scientists on the way that played a crucial part in development of this Electromagnetic theory such as Gauss, Faraday and Ampere. Then there were others such as Huygens, Fresnel and Young who worked on nature of light, which was not known to be an Electromagnetic wave at that time. Once the theory  of Electromagnetic wave propagation was in place there was rapid progress in many fields, particularly in wireless communications (wireless telegraph, radio, radar etc.).

Maxwell’s equations that were proposed in 1861 were initially quite circuitous and were not well accepted. But later on these equations were simplified into the form we now know by Oliver Heaviside. There are still two popular forms of the equations, the integral form and the differential form. We present the integral form of these equations in this article as it is more intuitive and is also easier to represent graphically. The differential form requires understanding of the concepts of divergence and curl and we skip it in this article. The main take away from these equations (presented below) is that a changing Electric field produces a Magnetic field and a changing Magnetic field produces an Electric field. Another important result is that magnetic monopoles do not exist (simply put a magnet, however small, always has a north and south pole).

Note:

1. The dot product with a line segment means that only that component of the field vector is effective that is along the line segment. On the other hand the dot product with a surface means that only that component is considered that is perpendicular to the surface (since the unit vector of a surface is perpendicular to the surface). It means that only those field components are considered that are going perpendicularly in or out of the surface.
2. For more on history of Maxwell equations visit IEEE Spectrum  and for a detailed explanation of the various forms of the Maxwell’s equations visit this page.
3. In modern Electromagnetic simulation software the differential form is preferred and the algorithm used is called Finite Difference Time Domain (FDTD). However, if the area of interest is quite large (with respect to the wavelength) then the FDTD method becomes prohibitively complex and another method known as Ray-Tracing is used. Please do check out the Ray-Tracing engine that we have developed. Ray-Tracing is becoming increasingly important in RF Planning of Telecom Networks.

# Knife Edge Diffraction Model

Diffraction is a phenomenon where electromagnetic waves (such as light waves) bend around corners to reach places which are otherwise not reachable i.e. not in the line of sight. In technical jargon such regions are also called shadowed regions (the term again drawn from the physics of light). This phenomenon can be explained by Huygen’s principle which states that as a plane wave propagates in a particular direction each new point along the wavefront is a source of secondary waves. This can be understood by looking at the following figure. However one peculiarity of this principle is that it is unable to explain why the new point source transmits only in the forward direction.

The electromagnetic field in the shadowed region can be calculated by combining vectorially the contributions of all of these secondary sources, which is not an easy task. Secondly, the geometry is usually much more complicated than shown in the above figure. For example consider a telecom tower transmitting electromagnetic waves from a rooftop and a pedestrian using a mobile phone at street level. The EM waves usually reach the receiver at street level after more than one diffraction (not to mention multiple reflections). However, an approximation that works well in most cases is called knife edge diffraction, which assumes a single sharp edge (an edge with a thickness much smaller than the wavelength) separates the transmitter and receiver.

The path loss due to diffraction in the knife edge model is controlled by the Fresnel Diffraction Parameter which measures how deep the receiver is within the shadowed region. A negative value for the parameter shows that the obstruction is below the line of sight and if the value is below -1 there is hardly any loss. A value of 0 (zero) means that the transmitter, receiver and tip of the obstruction are all in line and the Electric Field Strength is reduced by half or the power is reduced to one fourth of the value without the obstruction i.e. a loss of 6dB.  As the value of the Fresnel Diffraction Parameter increases on the positive side the path loss rapidly increases reaching a value of 27 dB for a parameter value of 5. Sometimes the exact calculation is not needed and only an approximate calculation, as proposed by Lee in 1985, is sufficient.

Fresnel Diffraction Parameter (v) is defined as:

v=h*sqrt(2*(d1+d2)/(lambda*d1*d2))

where

d1 is the distance between the transmitter and the obstruction along the line of sight

d2 is the distance between the receiver and the obstruction along the line of sight

h is the height of the obstruction above the line of sight

and lambda is the wavelength

The MATLAB codes used to generate the above plots are given below (approximate method followed by the exact method). Feel free to use them in your simulations and if you have a question drop us a comment.

```%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Calculation of the path loss based on the value of
% Fresnel Diffraction Parameter as proposed by Lee
% Lee W C Y Mobile Communications Engineering 1985
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all
close all

v=-5:0.01:5;

for n=1:length(v)

if v(n) <= -1
G(n)=0;
elseif v(n) <= 0
G(n)=20*log10(0.5-0.62*v(n));
elseif v(n) <= 1
G(n)=20*log10(0.5*exp(-0.95*v(n)));
elseif v(n) <= 2.4
G(n)=20*log10(0.4-sqrt(0.1184-(0.38-0.1*v(n))^2));
else
G(n)=20*log10(0.225/v(n));
end

end

plot(v, G, 'b')
xlabel('Fresnel Diffraction Parameter')
ylabel('Diffraction Loss (dB)')
```
```%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Exact calculation of the path loss (in dB)
% based on Fresnel Diffraction Parameter (v)
% T S Rappaport Wireless Communications P&P
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all
close all

v=-5:0.01:5;

for n=1:length(v)

v_vector=v(n):0.01:v(n)+100;

F(n)=((1+1i)/2)*sum(exp((-1i*pi*(v_vector).^2)/2));

end

F=abs(F)/(abs(F(1)));

plot(v, 20*log10(F),'r')
xlabel('Fresnel Diffraction Parameter')
ylabel('Diffraction Loss (dB)')
```

An interesting video explaining the phenomenon of diffraction.

# How to Find Point of Intersection of Two Lines

Finding the point of intersection of two lines has many important application such as in Ray-Tracing Simulation.  Two lines always intersect at some point unless they are absolutely parallel, like the rails of a railway track. We start with writing the equations of the two lines in slope-intercept form.

y1=b1+m1*x1

y2=b2+m2*x2

Here m1 and m2 are the slopes of the two lines and b1 and b2 are their y-intercepts. At the point of intesection y1=y2, so we have.

b1+m1*x1=b2+m2*x2

But at  the point of intersection x1=x2 as well, so replacing x1 and x2 with x we have.

b1+m1*x=b2+m2*x

or

b1-b2=-x*(m1-m2)

or

x=-(b1-b2)/(m1-m2)

Once the x-component of the point of intersection is found we can easily find the y-component by substituting x in any of the two line equations above.

y=b1+m1*x

In future posts we would like to discuss the cases of intersection of two surfaces and the intersection of two volumes.