Editing Fraunhofer diffraction (section)

==Examples==

In each of these examples, the aperture is illuminated by a monochromatic plane wave at normal incidence.

===Diffraction by a narrow rectangular slit===

[[Image:Single_Slit_Diffraction_(english).svg|right|thumb|Graph and image of single-slit diffraction]]

The width of the slit is {{mvar|W}}. The Fraunhofer diffraction pattern is shown in the image together with a plot of the intensity vs. angle {{mvar|θ}}.<ref>{{harvnb|Hecht|2002|loc=Figures 10.6(b) and 10.7(e)}}</ref> The pattern has maximum intensity at {{math|1=''θ'' = 0}}, and a series of peaks of decreasing intensity. Most of the diffracted light falls between the first minima. The angle, {{math|α}}, subtended by these two minima is given by:<ref>{{harvnb|Jenkins|White|1957|p=297}}</ref>

<math display="block"> \alpha \approx {\frac{2 \lambda}{W}} </math>

Thus, the smaller the aperture, the larger the angle {{math|α}} subtended by the diffraction bands.  The size of the central band at a distance {{math|''z''}} is given by

<math display="block">d_f = \frac {2 \lambda z}{W}</math>[[File:Difraction glass.jpg|thumb|225x225px|Diffraction glass with 300 lines per millimeter]]

For example, when a slit of width 0.5&nbsp;mm is illuminated by light of wavelength 0.6&nbsp;μm, and viewed at a distance of 1000&nbsp;mm, the width of the central band in the diffraction pattern is 2.4&nbsp;mm.

The fringes extend to infinity in the {{math|''y''}} direction since the slit and illumination also extend to infinity.

If {{math|W < λ}}, the intensity of the diffracted light does not fall to zero, and if {{math|D << λ}}, the diffracted wave is cylindrical.

====Semi-quantitative analysis of single-slit diffraction====
[[Image:single slit diagram.svg|100px|right|thumb|Geometry of single-slit diffraction]]
We can find the angle at which a first minimum is obtained in the diffracted light by the following reasoning. Consider the light diffracted at an angle {{math|θ}} where the distance {{math|''CD''}} is equal to the wavelength of the illuminating light. The width of the slit is the distance {{math|''AC''}}. The component of the wavelet emitted from the point A which is travelling in the {{math|θ}} direction is in [[phase (waves)#phase shift|anti-phase]] with the wave from the point {{math|''B''}} at middle of the slit, so that the net contribution at the angle {{math|θ}} from these two waves is zero.  The same applies to the points just below {{math|''A''}} and {{math|''B''}}, and so on.  Therefore, the amplitude of the total wave travelling in the direction {{math|θ}} is zero.  We have:

<math display="block">\theta_\text{min} \approx \frac {CD} {AC} = \frac{\lambda}{W}.</math>

The angle subtended by the first minima on either side of the centre is then, as above:

<math display="block">\alpha = 2 \theta_\text{min} = \frac{2\lambda}{W}.</math>

There is no such simple argument to enable us to find the maxima of the diffraction pattern.

==== Single-slit diffraction using Huygens' principle ====
[[File:Singleslithuygens.jpg|thumb|Continuous broadside array of point sources of length ''a''.]]
We can develop an expression for the far field of a continuous array of point sources of uniform amplitude and of the same phase. Let the array of length ''a'' be parallel to the y axis with its center at the origin as indicated in the figure to the right. Then the differential [[Electric field|field]] is:<ref name=":0">{{Cite book|url=https://books.google.com/books?id=NRxTAAAAMAAJ|title=Antennas for all applications|last1=Kraus|first1=John Daniel|last2=Marhefka|first2=Ronald J.|date=2002|publisher=McGraw-Hill|isbn=9780072321036|language=en}}</ref>

<math display="block">dE=\frac{A}{r_1}e^{i \omega [t-(r_1/c)]}dy=\frac{A}{r_1}e^{i(\omega t-\beta r_1)}dy</math>

where <math>\beta=\omega/c=2\pi /\lambda</math>. However <math>r_1=r-y\sin\theta</math> and integrating from <math>-a/2</math> to <math>a/2</math>,

<math display="block">E \simeq A' \int_{-a/2}^{a/2} e^{i\beta y \sin\theta} dy</math>
where <math>A' = \frac{Ae^{i(\omega t-\beta r)}}{r}</math>.

Integrating we then get
<math display="block">E = \frac{2A'}{\beta \sin \theta} \sin\left(\frac{\beta a}{2} \sin \theta\right)</math>

Letting <math>\psi^'=\beta a \sin \theta = \alpha_r \sin \theta</math> where the array length in radians is <math>a_r=\beta a=2\pi a/\lambda</math>, then,<ref name=":0" />
<math display="block">E= A' a \frac{\sin(\psi^'/2)}{\psi^'/2}</math>

===Diffraction by a rectangular aperture===

[[File:Rectangular diffraction.jpg|right|200px|thumbnail|Computer simulation of Fraunhofer diffraction by a rectangular aperture]]The form of the diffraction pattern given by a rectangular aperture is shown in the figure on the right (or above, in tablet format).<ref>{{harvnb|Born|Wolf|1999|loc=Figure 8.10}}</ref> There is a central semi-rectangular peak, with a series of horizontal and vertical fringes.  The dimensions of the central band are related to the dimensions of the slit by the same relationship as for a single slit so that the larger dimension in the diffracted image corresponds to the smaller dimension in the slit.  The spacing of the fringes is also inversely proportional to the slit dimension.

If the illuminating beam does not illuminate the whole vertical length of the slit, the spacing of the vertical fringes is determined by the dimensions of the illuminating beam.  Close examination of the double-slit diffraction pattern below shows that there are very fine horizontal diffraction fringes above and below the main spot, as well as the more obvious horizontal fringes.

===Diffraction by a circular aperture===
[[File:Airy-pattern2.jpg|thumb|150px|right|Computer simulation of the Airy diffraction pattern]]The diffraction pattern given by a circular aperture is shown in the figure on the right.<ref>{{harvnb|Born|Wolf|1999|loc=Figure 8.12}}</ref> This is known as the [[Airy Disk|Airy diffraction pattern]]. It can be seen that most of the light is in the central disk. The angle subtended by this disk, known as the Airy disk, is
<math display="block"> \alpha \approx \frac {1.22 \lambda} {W}</math>
where {{math|''W''}} is the diameter of the aperture.

The Airy disk can be an important parameter in [[Angular resolution|limiting the ability]] of an imaging system to resolve closely located objects.

===Diffraction by an aperture with a Gaussian profile===

[[Image:Exp squared function.svg|right|200px|thumb|Intensity of a plane wave diffracted through an aperture with a Gaussian profile]] The diffraction pattern obtained given by an aperture with a [[Gaussian function|Gaussian]] profile, for example, a photographic slide whose [[Transmittance|transmissivity]] has a Gaussian variation is also a Gaussian function. The form of the function is plotted on the right (above, for a tablet), and it can be seen that, unlike the diffraction patterns produced by rectangular or circular apertures, it has no secondary rings.<ref>{{harvnb|Hecht|2002|loc=Figure 11.33}}</ref> This technique can be used in a process called [[apodization]]—the aperture is covered by a Gaussian filter, giving a diffraction pattern with no secondary rings.

The output profile of a single mode laser beam may have a [[Gaussian beam|Gaussian]] intensity profile and the diffraction equation can be used to show that it maintains that profile however far away it propagates from the source.<ref>{{harvnb|Hecht|2002|loc=Figure 13.14}}</ref>

===Diffraction by a double slit===
[[File:SodiumD two double slits 2.jpg|thumb|200px|Double-slit fringes with sodium light illumination]]
In the [[double-slit experiment]], the two slits are illuminated by a single light beam.  If the width of the slits is small enough (less than the wavelength of the light), the slits diffract the light into cylindrical waves. These two cylindrical wavefronts are superimposed, and the amplitude, and therefore the intensity, at any point in the combined wavefronts depends on both the magnitude and the phase of the two wavefronts.<ref>{{harvnb|Born|Wolf|1999|loc=Figure 7.4}}</ref> These fringes are often known as [[Young's interference experiment|Young's fringes]].

The angular spacing of the fringes is given by
<math display="block">\theta_\text{f} = \lambda/d.</math>

The spacing of the fringes at a distance {{mvar|z}} from the slits is given by<ref>{{harvnb|Hecht|2002|loc=eq. (9.30).}}</ref>
<math display="block">w_\text{f} = z \theta_f = z \lambda/d,</math>
where {{mvar|d}} is the separation of the slits.

The fringes in the picture were obtained using the yellow light from a sodium light (wavelength = 589&nbsp;nm), with slits separated by 0.25&nbsp;mm, and projected directly onto the image plane of a digital camera.

Double-slit interference fringes can be observed by cutting two slits in a piece of card, illuminating with a laser pointer, and observing the diffracted light at a distance of 1&nbsp;m.  If the slit separation is 0.5&nbsp;mm, and the wavelength of the laser is 600&nbsp;nm, then the spacing of the fringes viewed at a distance of 1&nbsp;m would be 1.2&nbsp;mm.

====Semi-quantitative explanation of double-slit fringes====

[[Image:Double slit.svg|thumbnail|right|300px|Geometry for far-field fringes]]
The difference in phase between the two waves is determined by the difference in the distance travelled by the two waves.

If the viewing distance is large compared with the separation of the slits (the [[far field]]), the phase difference can be found using the geometry shown in the figure.  The path difference between two waves travelling at an angle {{mvar|θ}} is given by
<math display="block">d \sin \theta \approx d \theta.</math>

When the two waves are in phase, i.e. the path difference is equal to an integral number of wavelengths, the summed amplitude, and therefore the summed intensity is maximal, and when they are in anti-phase, i.e. the path difference is equal to half a wavelength, one and a half wavelengths, etc., then the two waves cancel, and the summed intensity is zero.  This effect is known as [[Interference (optics)|interference]].

The interference fringe maxima occur at angles
<math display="block">d \theta_n = n \lambda,\quad n = 0, \pm 1, \pm 2, \ldots</math>
where {{mvar|λ}} is the [[wavelength]] of the light. The angular spacing of the fringes is given by
<math display="block">\theta_\text{f} \approx \lambda/d.</math>

When the distance between the slits and the viewing plane is {{math|''z''}}, the spacing of the fringes is equal to {{math|''zθ''}} and is the same as above:
<math display="block">w = z\lambda / d.</math>

===Diffraction by a grating===

[[File:Diffraction of laser beam by grating.jpeg|thumb|150px|Diffraction of a laser beam by a grating]] A grating is defined in Born and Wolf as "any arrangement which imposes on an incident wave a periodic variation of amplitude or phase, or both".

A grating whose elements are separated by {{math|''S''}} diffracts a normally incident beam of light into a set of beams, at angles {{math|''θ''<sub>''n''</sub>}} given by:<ref>{{Cite book | last = Longhurst | first = R. S. | title = Geometrical and Physical Optics | edition = 2nd | year = 1967 | publisher = Longmans | location = London | at = eq.(12.1)}}</ref>

<math display="block">~ \sin \theta_n = \frac{n \lambda} {S}, \quad n = 0, \pm 1, \pm 2, \ldots </math>

This is known as the [[Diffraction grating|grating equation]]. The finer the grating spacing, the greater the angular separation of the diffracted beams.

If the light is incident at an angle {{math|θ<sub>0</sub>}}, the grating equation is:
<math display="block">\sin \theta_n = \frac {n \lambda} {S} + \sin \theta_0, \quad n=0, \pm 1, \pm 2, \ldots </math>

The detailed structure of the repeating pattern determines the form of the individual diffracted beams, as well as their relative intensity while the grating spacing always determines the angles of the diffracted beams.

The image on the right shows a laser beam diffracted by a grating into {{math|''n''}} = 0, and ±1 beams.  The angles of the first order beams are about 20°; if we assume the wavelength of the laser beam is 600&nbsp;nm, we can infer that the grating spacing is about 1.8&nbsp;μm.

====Semi-quantitative explanation====

[[Image:Beugungsgitter.svg|200px|thumb|right]] A simple grating consists of a series of slits in a screen. If the light travelling at an angle {{math|θ}} from each slit has a path difference of one wavelength with respect to the adjacent slit, all these waves will add together, so that the maximum intensity of the diffracted light is obtained when:
<math display="block">W \sin \theta = n \lambda, \quad n=0, \pm 1, \pm 2, \ldots </math>

This is the same relationship that is given above.