# Fourier optics

The Fourier optics (according to Jean Baptiste Joseph Fourier ) is a portion of the optics , in which the propagation of light with the aid of Fourier analysis is examined. Fourier optics take into account the wave nature of light, but neglect z. B. the polarization .

## background

The basis of Fourier optics is the finding that the Fraunhofer diffraction pattern corresponds to the Fourier transform of the diffracting object.

If coherent light with the spatial amplitude distribution falls on a structure with the spatial transmission distribution , the field distribution is immediately behind the diffractive structure: ${\ displaystyle E_ {e}}$${\ displaystyle \ tau}$

${\ displaystyle E_ {t} = E_ {e} \ cdot \ tau.}$

In the far field of the structure, the following applies to the amplitude distribution:

${\ displaystyle E (x, y) = A (x, y, z_ {0}) \ cdot \ int \ limits _ {- \ infty} ^ {\ infty} E_ {t} (x ', y') \ cdot \ operatorname {e} ^ {- i2 \ pi (xx '+ yy') / (\ lambda z_ {0})} \ mathrm {d} x '\ mathrm {d} y'.}$

It is

• ${\ displaystyle z_ {0}}$ the distance from the diffractive structure
• ${\ displaystyle x, y}$ the transverse coordinates
• ${\ displaystyle A}$ a phase factor.
• ${\ displaystyle \ lambda}$ the wavelength

The spatial frequencies are defined analogously to the frequency in the temporal Fourier transformation :

{\ displaystyle {\ begin {aligned} \ nu _ {x} &: = {\ frac {x} {\ lambda \ cdot z_ {0}}} \\\ nu _ {y} &: = {\ frac { y} {\ lambda \ cdot z_ {0}}}, \ end {aligned}}}

it follows

${\ displaystyle \ Rightarrow E (x, y) = A (x, y, z_ {0}) \ cdot \ int \ limits _ {- \ infty} ^ {\ infty} E_ {t} (x ', y' ) \ operatorname {e} ^ {- i2 \ pi (\ nu _ {x} x '+ \ nu _ {y} y')} \ mathrm {d} x '\ mathrm {d} y'.}$

The far field is given by the two-dimensional Fourier transform of the field immediately behind the diffractive structure: ${\ displaystyle {\ mathcal {F}}}$${\ displaystyle E_ {t}}$

${\ displaystyle E = A \ cdot {\ mathcal {F}} \ left [E_ {t} \ right] \ left (\ nu _ {x}, \ nu _ {y} \ right).}$

### Importance of spatial frequencies

A ray from the point in the observation plane to the point in the plane of the diffractive structure includes the following angles with the axis: ${\ displaystyle (x, y, z_ {0})}$${\ displaystyle (0,0,0)}$${\ displaystyle z}$

${\ displaystyle \ tan (\ alpha) = {\ frac {x} {z_ {0}}} = \ lambda \ cdot \ nu _ {x} \ qquad \ tan (\ beta) = {\ frac {y} { z_ {0}}} = \ lambda \ cdot \ nu _ {y}.}$

For angles that are not too large ( i.e. not too large ) it follows from this ( small-angle approximation ): ${\ displaystyle x, y}$

${\ displaystyle \ alpha \ approx {\ frac {x} {z_ {0}}} = \ lambda \ cdot \ nu _ {x} \ qquad \ beta \ approx {\ frac {y} {z_ {0}}} = \ lambda \ cdot \ nu _ {y}.}$

Light that is close to the optical axis in the far field corresponds to low spatial frequencies, while light located further out belongs to high spatial frequencies.

Fine structures in the object, i.e. those that change quickly in space, belong to high spatial frequencies; accordingly, coarser structures represent lower spatial frequencies.