Diffraction Grating Hyperphysics

 

As the screen is far away enough from the grating, we approximate all the factors (sqrt{l_i}) of the decrease of the magnitude with a constant value (sqrt{L}).

  1. Diffraction Grating Hyperphysics Meaning
  2. Diffraction Grating Hyperphysics Examples
  3. Diffraction Grating Formula
  4. Diffraction Grating Hyperphysics

The electric field of the wave coming from the 1st, 2nd, to nth slit at a given point at the screen that is at a distance (l_1,l_2) to (l_n), respectively, from these slits, is:

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  • DIFFRACTION GRATING — component of optical devices consisting of a surface ruled with close, equidistant, and parallel lines for the purpose of.
Diffraction grating hyperphysics meaning[ begin{eqnarray} E_1 &=& frac{mathcal{E}_0}{sqrt{L}} e^{i(omega t - kl_1)}, E_2 &=& frac{mathcal{E}_0}{sqrt{L}} e^{i(omega t - kl_2)}, &vdots& E_n &=& frac{mathcal{E}_0}{sqrt{L}} e^{i(omega t - kl_n)}. end{eqnarray} ]

The resulting field at a given point at the screen is directly the sum of these fields, because they are all in the same polarization state:

[E = E_1 + E_2 + ldots + E_n = frac{mathcal{E}_0}{sqrt{L}} left( e^{i(omega t - kl_1)} + e^{i(omega t - kl_2)} + ldots + e^{i(omega t - kl_n)} right).]

We factor out the exponents (phases) of time and of the distance to the first slit:

[=frac{mathcal{E}_0}{sqrt{L}} e^{i(omega t - kl_1)} big(e^{-ik(overbrace{l_1-l_1}^{mathrm{def.,}Delta l_1})} + e^{-ik(overbrace{l_2-l_1}^{mathrm{def.,}Delta l_2})} + ldots + e^{-ik(overbrace{l_n-l_1}^{mathrm{def.,}Delta l_n})} big)]Hyperphysics and we symbolically sum the path differences: [= frac{mathcal{E}_0}{sqrt{L}} e^{i(omega t - kl_1)} big(e^{-ik Delta l_1} + e^{-ik Delta l_2} + ldots + e^{-ik Delta l_n} big)=][= frac{mathcal{E}_0}{sqrt{L}} e^{i(omega t - kl_1)} sum_{j=1}^{n} e^{-ik Delta l_j}. tag{1}]Diffraction

Let us now focus on the result of the summation. We express the path difference of the (j)th wave with respect to the first slit using a multiple of the grating constant (b) an angle (alpha).

We draw perpendiculars from the first slit to the wave paths. For a far-enough screen, we can consider the paths from different slits to be parallel and all the angles (alpha) to be the same.

It then holds that

Diffraction Grating Hyperphysics Meaning

[sin alpha = frac{Delta l_j}{(j-1) cdot b}]

and the path difference for the (j)th slit is:

[Delta l_j = (j-1) , b , sinalpha.]

Upon substituting into Equation (1), we get

[E = frac{mathcal{E}_0}{sqrt{L}} e^{i(omega t - kl_1)} sum_{j=1}^{n} e^{-ik (j-1)bsinalpha}.]

We can see, that we are summing the (n) first terms of a geometric series with the first term (1) and the common ratio (e^{-ikbsinalpha}). We perform the summation:

[E = frac{mathcal{E}_0}{sqrt{L}} e^{i(omega t - kl_1)} sum_{j=1}^{n} big(underbrace{e^{-ik bsinalpha}}_{mathrm{kvocient}}big)^{j-1} = frac{mathcal{E}_0}{sqrt{L}} e^{i(omega t - kl_1)} frac{left(e^{-ikbsinalpha}right)^n - 1 }{e^{-ikbsinalpha} - 1}.]

And we get the resulting electric field after a simple arrangement:

Diffraction Grating Hyperphysics Examples

[E = frac{mathcal{E}_0}{sqrt{L}} e^{i(omega t - kl_1)} frac{e^{-inkbsinalpha} - 1 }{e^{-ikbsinalpha} - 1}.]

Diffraction Grating Formula

The complex conjugate of the resulting electric field is then:

Diffraction Grating Hyperphysics

[E^star = frac{mathcal{E}_0}{sqrt{L}} e^{-i(omega t - kl_1)} frac{e^{inkbsinalpha} - 1 }{e^{ikbsinalpha} - 1}.]