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I am trying to derive the intensity variation function for a single slit diffraction. Diagram

Sorry for the poor diagram...

So I decided to take the amplitudes of the waves originating from the slit on the left (wherein the variable that denotes distance within the slit is $l$) and integrate the amplitudes over the entire slit width, taking some point at a distance $x$ on the screen to achieve the resultant amplitude of the waves that strike the screen. With this function, I decided I would use the standard expression for intensity (i.e. $I=\kappa A^2)$

The amplitude for a wave originating from a point on the slit should be: $$ y=a\sin{kr}$$ where $r$ is the distance between the point of origin on the slit and point of contact on the screen (and $k$ is the angular wave-number). So: $$ r^2=D^2+(x+l)^2$$ and on approximating: $$ r\approx D+\frac{1}{2D}(x+l)^2$$ So I took the amplitude function (for the screen) as $A(x)$ and: $$ A(x)=a\int_{-l/2}^{l/2}\sin{kD+\frac{k}{2D}(x+l)^2} dl$$ substituting $k(x+l)/2D=u$ (ignoring limits for now): $$ A(x)=a\sqrt{\frac{2D}{k}}(\sin{kD}\int_{l_1}^{l_2}\cos{u^2}du+\cos{kD}\int_{l_1}^{l_2}\sin{u^2}du)$$ I looked these integrals up so I know that they are Fresnel Integrals, but more importantly that they are transcendental functions.

So my questions are:

  1. Are my assumptions flawed?
  2. Is there a flaw somewhere in the procedure?
  3. If what I've done is correct, how shall I proceed?
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There is a flaw in your procedure. You appear to be integrating over x (which would mean that A(x) doesn't depend on x), but what you want is to express how r varies depending on where the light source is, along the slit. This is what StarDrop's answer is doing. – Dr Chuck Oct 27 at 17:06
@DrChuck : So sorry... I missed the $dl$ (I have edited it in the question). As you can see, I was integrating over $l$, not $x$, to find the amplitude at a point $x$ on the screen. So, $x$ is taken to be constant for the procedure. – Prish Chakraborty Oct 27 at 20:40

1 Answer 1

Try using this method.

enter image description here

To study diffraction of light, laser light is passed through a narrow single slit and the diffraction pattern is formed on a distant screen. An imaginary reference line is drawn perpendicularly from the center of the slit out to the screen (see Figure 3), which is a distance L away. The intensity variation of the diffraction pattern can then be measured accurately as a function of the distance y from the reference line. In the theoretical description of the diffraction pattern, however, it is more convenient to quantify the light intensity as a function of the sine of the angle θ defined accordingly by

$sin θ = y/\sqrt{y2+L2}$

The theory of diffraction predicts that the spatial pattern of light intensity on the viewing screen by a light wave passing through a single rectangular-shaped slit is given by

enter image description here

(4) where I0 is the light intensity at θ = 0◦ and the quantities in parentheses are in radians.

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Thanks for the input, but please note that I wished to arrive at the expression you've mentioned (or something similar) through a different approach, namely the one I've detailed in the question. – Prish Chakraborty Oct 27 at 15:08

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