Ray Airy

Airy beam ( English Airy beam ) - undiffracting waveform , which manifests itself in the form of bending as the beam propagates.

Content

Physical Description

In cross section, the Airy ray is the region where the main intensity falls, the brightness of neighboring regions decays sequentially, converging to zero at infinity. In practice, the beam is truncated to obtain finite values in a limited area.

When propagating, the Airy beam does not undergo diffraction , that is, does not spread. This beam is characterized by free acceleration : as it propagates, it deviates from the original direction, forming an arc of a parabola .

History

The term “Airy ray” comes from the Airy integral , introduced in 1838 by Sir George Biddel Airy to explain optical caustics , such as those that appear as rainbows ^[1] .

The existence of an Airy ray was first theoretically hypothesized by Michael Berry and in 1979. They demonstrated a solution in the form of a non - expandable Airy wave packet for the Schrödinger equation ^[2] .

For the first time, researchers at the University of Central Florida in 2007 created and observed the Airy ray in the form of one- and two-dimensional configurations. The team included Georgios Siviloglou, John Broki, Aristide Dogariu and Dimitrios Christodoulides ( Georgios Siviloglou, John Broky, Aristide Dogariu, and Demetrios Christodoulides ) ^[1] .

In the one-dimensional case, the Airy ray is the only waveform-preserving accelerating solution of the Schrödinger equation for a free particle (the same is true for two-dimensional wave optics of paraxial rays). However, in two dimensions (or for three-dimensional paraxial optical systems) two solutions are possible: two-dimensional Airy rays and accelerated parabolic rays ^[3] .

Mathematical Description

Schrödinger equation in the absence of potential :

i{\frac {\partial \Phi }{\partial \xi }}+{\frac {1}{2}}{\frac {\partial ^{2}\Phi }{\partial \,s^{2}}}=0

{\ displaystyle i {\ frac {\ partial \ Phi} {\ partial \ xi}} + {\ frac {1} {2}} {\ frac {\ partial ^ {2} \ Phi} {\ partial \, s ^ {2}}} = 0}

has the following non-dispersive Airy solution ^[4] :

$\Phi (\xi ,\,s)=\operatorname {Ai} (\,s-(\xi /2)^{2})\exp(i(\,s\xi /2)-i(\xi ^{3}/12)),$ ${\ displaystyle \ Phi (\ xi, \, s) = \ operatorname {Ai} (\, s - (\ xi / 2) ^ {2}) \ exp (i (\, s \ xi / 2) -i (\ xi ^ {3} / 12)),}$

Where

Ai is the Airy function ;
$\Phi$ ${\ displaystyle \ Phi}$ - envelope of the electric field ;
$s=x/x_{0}$ ${\ displaystyle s = x / x_ {0}}$ - dimensionless transverse coordinate;
$x_{0}$ ${\ displaystyle x_ {0}}$ - arbitrary transverse scale;
$\xi =z/kx_{0}^{2}$ ${\ displaystyle \ xi = z / kx_ {0} ^ {2}}$ - normalized propagation distance (longitudinal coordinate);
$k=2\pi \,n/\lambda _{0}$ ${\ displaystyle k = 2 \ pi \, n / \ lambda _ {0}}$

Experimental observation

Georgios Civiloglu and co-authors successfully created the Airy optical beam in 2007 . To obtain an Airy propagation, a beam with a Gaussian distribution was modulated by a spatial light modulator . The result was recorded on a CCD camera ^[1] .

In 2013, the Airy electron beam was first obtained ^[5] .

Application

Researchers at St. Andrews University used the Airy beam to control small particles by moving them along lines and around corners. This may find application in microfluidics and cell biology ^[6] .

Notes

↑ ¹ ² ³ “Scientists make first observation of Airy optical beams”
↑ M.V. Berry , N. Balage , Nonspreading wave packets , 47 (3), 1979, pp. 264-267
↑ MA Bandres. Accelerating parabolic beams. Opt. Lett. 33, 1678-1680 (2008).
↑ - "Observation of Accelerating Airy Rays" (English)
↑ Airy electron beam first received. Compulent, February 25, 2013
↑ PhysOrg.com: “Light throws the ball along a curve,” September 29, 2008. (eng.)

[physorg.com-1] ¹ ² ³ “Scientists make first observation of Airy optical beams”

[2] M.V. Berry , N. Balage , Nonspreading wave packets , 47 (3), 1979, pp. 264-267

[ApA-3] MA Bandres. Accelerating parabolic beams. Opt. Lett. 33, 1678-1680 (2008).

[4] - "Observation of Accelerating Airy Rays" (English)

[5] Airy electron beam first received. Compulent, February 25, 2013

[6] PhysOrg.com: “Light throws the ball along a curve,” September 29, 2008. (eng.)