PX275 - I5 - different path lengths

different path lengths lead to phase differences
considering two pinholes, hole 1 at the origin, and hole 2 at position, $\vec{y}$
at a point $\vec{x}$ on the detector:

\begin{matrix} l_{1} = | \vec{x} - {\vec{y}}^{1} | = | \vec{x} | \\ l_{2} = | \vec{x} - {\vec{y}}^{2} | = | \vec{x} - \vec{y} | \\ Δ ϕ = (l_{2} - l_{1}) k = k (| \vec{x} - \vec{y} | - | \vec{x} |) \end{matrix}

where, $Δ ϕ$ is the phase difference

\begin{aligned} | \vec{x} - \vec{y} | & = \sqrt{(\vec{x} - \vec{y}) (\vec{x} - \vec{y})} \\ = \sqrt{| {\vec{x}}^{2} | - 2 \vec{x} \cdot \vec{y} + | \vec{y} |^{2}} \\ = | \vec{x} | \sqrt{1 - \frac{2 \vec{x} \cdot \vec{y}}{| \vec{x} |^{2}} + \frac{| \vec{y} |^{2}}{| \vec{x} |^{2}}} \end{aligned}

considering for far field regime: the detector distance, $D ≫$ size of the aperture/detector screen
this is also called fraunhofer diffraction, as opposed to the near field regime - fresnel diffraction

∴ | \vec{x} | ≫ | \vec{y} | ⟹ | \vec{x} - \vec{y} | ≃ | \vec{x} | \sqrt{1 - ϵ}

where, $ϵ = 2 \vec{x} \cdot \vec{y} / | \vec{x} |^{2}$

using taylor expansion:

\begin{matrix} \sqrt{1 - ϵ} ≃ 1 - \frac{1}{2} ϵ + 0 (ϵ^{2}) + \dots \\ ∴ | \vec{x} - \vec{y} | ≃ | \vec{x} | (1 - \frac{\vec{x} \cdot \vec{y}}{| \vec{x} |^{2}}) \end{matrix}

∴ Δ ϕ = k (| \vec{x} | (1 - \frac{\vec{x} \cdot \vec{y}}{| \vec{x} |^{2}}) - | x |) = - k \frac{\vec{x} \cdot \vec{y}}{| \vec{x} |^{2}}

defining a wavevector:

\vec{k} = k \hat{x} = k \frac{\vec{x}}{| \vec{x} |}

this is also known as the 'scattering wavevector', and points towards the observation point

∴ Δ ϕ = - \vec{k} \cdot \vec{y}

for a single point source:

u (\vec{x}, t) = \frac{A}{| \vec{x} - \vec{y} |} \exp (i k (| \vec{x} - \vec{y} | - c t))

k | \vec{x} - \vec{y} | = k | \vec{x} | + Δ ϕ = k | \vec{x} | - \vec{k} \cdot \vec{y}

for a superposition of $N$ point sources:

\begin{aligned} u (\vec{x}, t) & = \sum_{j = 1}^{N} \frac{A^{j}}{| \vec{x} - {\vec{y}}^{j} |} \exp (i k (| \vec{x} - {\vec{y}}^{j} | - c t)) \\ = \sum_{j = 1}^{N} \frac{A^{j}}{| \vec{x} - {\vec{y}}^{j} |} \exp (i k | \vec{x} | - i \vec{k} \cdot {\vec{y}}^{j} - i k c t) \\ = \sum_{j = 1}^{N} \frac{A^{j}}{| \vec{x} - {\vec{y}}^{j} |} \exp (i k (| \vec{x} | - c t) - i \vec{k} \cdot {\vec{y}}^{j}) \end{aligned}

in far-field regime: $| \vec{x} - {\vec{y}}^{j} | ≃ D$ , so, $\exp (i k (| \vec{x} | - c t))$ is the same for all sources

∴ u (\vec{x}, t) = \frac{1}{D} \exp (i k (| \vec{x} | - c t)) \sum_{j = 1}^{N} A^{j} \exp (- i \vec{k} \cdot {\vec{y}}^{j})

this can be used to model various apertures
considering an aperture function, $a (\vec{y})$ , that describes the location of the pinholes, such that $a (\vec{y}) = 0$ except at the location with holes
eg: a set of pinholes, each described by a $δ$ function:

a (\vec{y}) = \sum_{j = 1}^{N} A^{j} δ (\vec{y} - {\vec{y}}^{j})

on a centred circular aperture with radius, $d$ :

a (\vec{y}) = {\begin{cases} 1 & if | \vec{y} | < d, \\ 0 & if | \vec{y} | \geq d . \end{cases}

the combined source contribution is an integral over $a (\vec{y})$

u (\vec{x}, t) = \frac{1}{D} \exp (i k (| \vec{x} | - c t)) \iint \exp (- i \vec{k} \cdot \vec{y}) a (\vec{y}) d y_{1} d y_{2}

the above integral is a 2D fourier transform of $a (\vec{y})$
thus, the spatial interference pattern for a given aperture function, $a (\vec{y})$ , is given by the fourier transform of the aperture function:

u (\vec{x}, t) \propto \tilde{a} (\vec{k})

the corresponding light intensity, $I (x_{1}, x_{2}) \propto | u |^{2} \propto | \tilde{a} (k) |^{2}$