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Definition/Summary
This entry describes diffraction of a wave when it passes through a single narrow slit.
Equations
The far-field (Fraunhofer) diffraction pattern has a power per area (irradiance) at an angle [itex]\theta[/itex] from a single slit of width [itex] d[/itex], for wavelength [itex]\lambda[/itex] and wavenumber [itex]k\ =\ 2\pi/\lambda[/itex] of:
[tex]I(\theta)\ =\ \left( \frac{\sin \beta}{\beta}\right)^2\,I(0)[/tex]
where:
[tex]\beta\ \equiv\ \frac{\pi d}{\lambda} \ \sin\theta\ =\ \frac{k d}{2} \ \sin\theta[/tex]
which for very small angles is approximately:
[tex]\beta \ \approx \ \frac{\pi d}{\lambda} \ \theta \ = \ \frac{kd}{2} \ \theta[/tex]The diffraction minima (dark fringes) occur when
[tex]\beta \ = \ n \pi, \ \ n \ = \ \pm 1, \ \pm 2, \ \pm 3, \ ...[/tex]
or, for small angles,
[tex]\theta \ \approx \ n \lambda / d, \ \ n \ = \ \pm 1, \ \pm 2, \ \pm 3, \ ...[/tex]
Note that n=0 corresponds to the central maximum, not a minimum.
Extended explanation
Definition of terms
* This entry is from our old Library feature, and was originally created by Redbelly98
This entry describes diffraction of a wave when it passes through a single narrow slit.
Equations
The far-field (Fraunhofer) diffraction pattern has a power per area (irradiance) at an angle [itex]\theta[/itex] from a single slit of width [itex] d[/itex], for wavelength [itex]\lambda[/itex] and wavenumber [itex]k\ =\ 2\pi/\lambda[/itex] of:
[tex]I(\theta)\ =\ \left( \frac{\sin \beta}{\beta}\right)^2\,I(0)[/tex]
where:
[tex]\beta\ \equiv\ \frac{\pi d}{\lambda} \ \sin\theta\ =\ \frac{k d}{2} \ \sin\theta[/tex]
which for very small angles is approximately:
[tex]\beta \ \approx \ \frac{\pi d}{\lambda} \ \theta \ = \ \frac{kd}{2} \ \theta[/tex]The diffraction minima (dark fringes) occur when
[tex]\beta \ = \ n \pi, \ \ n \ = \ \pm 1, \ \pm 2, \ \pm 3, \ ...[/tex]
or, for small angles,
[tex]\theta \ \approx \ n \lambda / d, \ \ n \ = \ \pm 1, \ \pm 2, \ \pm 3, \ ...[/tex]
Note that n=0 corresponds to the central maximum, not a minimum.
Extended explanation
Definition of terms
I = irradiance of the wave, with SI units of W/m2
I(0) = the irradiance at θ=0
d = the slit width
λ = the wavelength of the wave
k = 2π/λ
θ = the angle at which the irradiance is evaluated
I(0) = the irradiance at θ=0
d = the slit width
λ = the wavelength of the wave
k = 2π/λ
θ = the angle at which the irradiance is evaluated
* This entry is from our old Library feature, and was originally created by Redbelly98
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