Bessel beam

Cross section through the intensity distribution of a Bessel beam

A Bessel beam (after the German astronomer , mathematician , geodesist and physicist Friedrich Wilhelm Bessel ) describes a special, ideal form of electromagnetic waves in wave optics . One of their most important application- related properties is that they are not diffractive : their shape - unlike Gaussian rays - does not change during propagation . A Bessel ray always propagates at less than the speed of light .

Mathematically, a Bessel ray is a set of solutions to the paraxial Helmholtz equation , which describes the shape of paraxial rays in wave optics.

Bessel beams were theoretically constructed in 1987 by Jim Durnin and demonstrated experimentally by Durnin and Joseph H. Eberly .

Mathematical description

In paraxial optics, rays of light that are parallel to the -axis are described as electromagnetic waves in the form: ${\displaystyle z}$

${\displaystyle U({\vec {r}},t)=A(x,y,z)\cdot \mathrm {e} ^{\mathrm {i} \,k_{z}\,z}\cdot \mathrm {e} ^{\mathrm {i} \,\omega \,t},}$

in which

• ${\displaystyle {\vec {r}}=(x,y,z)}$ the place and
• ${\displaystyle t}$ the time is.
• The amplitude function should depend at most weakly on, so that the wave along the z-axis is approximately periodic with the wavelength . ${\displaystyle A(x,y,z)}$ ${\displaystyle z}$ ${\displaystyle \lambda _{z}=2\pi /k_{z}}$
• For the oscillation angular frequency must apply . ${\displaystyle \omega \geq k_{z}\,c}$
• For the size of that in a plane wave that to the angular frequency corresponding wave number is so true . The size is called the transverse wavenumber. ${\displaystyle k={\frac {\omega }{c}}}$ ${\displaystyle \omega }$ ${\displaystyle k\geq k_{z}}$
  ${\displaystyle k_{T}={\sqrt {k^{2}-k_{z}^{2}}}}$

Is the amplitude even completely independent of ( dispersion- free beam): ${\displaystyle z}$

${\displaystyle A=A(x,y)\neq f(z)}$,

so the paraxial Helmholtz differential equation must satisfy: ${\displaystyle A(x,y)}$

${\displaystyle \nabla _{T}^{2}\,A(x,y)+k_{T}^{2}\,A(x,y)=0}$

with the Laplace operator restricted to the xy plane . The following applies in polar coordinates : ${\displaystyle \nabla _{T}^{2}={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}}$ ${\displaystyle x=\rho \cdot \cos(\phi ),\ \ y=\rho \cdot \sin(\phi )}$

${\displaystyle \nabla _{T}^{2}={\frac {\partial ^{2}}{\partial \rho ^{2}}}+{\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}}{\partial \phi ^{2}}}.}$

The solution to the above The differential equation, assuming cylindrical symmetry, gives :

${\displaystyle \Rightarrow A(x,y)=A\cdot J_{m}(k_{T}\cdot \rho )\cdot \mathrm {e} ^{\mathrm {i} \,m\,\phi },\ \ m=0,\pm 1,\pm 2,...}$

The Bessel function is the 1st type of order . ${\displaystyle J_{m}(\cdot )}$ ${\displaystyle m}$

The rotationally symmetrical special case is usually referred to simply as a Bessel beam . ${\displaystyle m=0}$

In the limit case, the result is a plane wave with the wave vector and . ${\displaystyle k_{T}\rightarrow 0}$ ${\displaystyle A(x,y)=const}$ ${\displaystyle {\vec {k}}=(0,0,k_{z})}$ ${\displaystyle k_{z}=\omega /c}$

properties

Comparison of the radial intensity of a Gaussian beam (red) and a Bessel beam with m = 0 (blue). In the upper diagram the 1 / e² width of the central peak and in the lower diagram the intensity integrated over the plot area correspond.

profile

The lateral amplitude and thus also the lateral intensity distribution

${\displaystyle I_{m}(\rho ,\phi ,z)=|A_{0}|^{2}\cdot J_{m}^{2}(k_{T}\cdot \rho )}$

do not depend on the position in the direction of propagation: ${\displaystyle z}$

${\displaystyle I_{m}\neq f(z)}$

Therefore, in contrast to Gaussian rays, for example, their width does not change during propagation. One speaks therefore of non-diffractive rays. This also means that Bessel rays do not have a focus in the sense of a point of highest intensity along the direction of propagation.

Bessel rays are also called "self-healing" because they can be partially disturbed or blocked at one point on the axis of propagation, for example by a scattering center, but regain their shape later in the direction of propagation.

The intensity in the radial direction drops significantly less with Bessel rays than with Gauss rays, cf. adjacent images: ${\displaystyle \rho }$

${\displaystyle I_{0}^{\text{Bessel}}(\rho )\propto J_{0}^{2}(k_{T}\cdot \rho )\approx {\frac {1}{\rho }}\cdot \cos ^{2}(k_{T}\rho -\pi /4)}$

However, for Gaussian rays:

${\displaystyle I_{0}^{\text{Gauß}}(\rho ,z)\propto \exp \left(-{\frac {2\rho ^{2}}{W(z)}}\right)}$

Because of the drop, an ideal Bessel beam also contains an infinite amount of energy, since the integral over the intensity diverges in the radial direction, i.e. H. does not approach a maximum. This is one reason why ideal Bessel rays cannot be realized in practice. ${\displaystyle 1/\rho }$

As can also be seen from the figure, Bessel rays have clear secondary maxima (also in comparison to an Airy disk ), but have a significantly narrower main maximum with the same energy content.

speed

The phase velocity in the direction of propagation is: ${\displaystyle z}$

${\displaystyle c_{\text{Phase}}={\frac {\omega }{k_{z}}}}$

and therefore greater  (!) than the speed of light : ${\displaystyle c}$

${\displaystyle \Rightarrow c_{\text{Phase}}>c={\frac {\omega }{k}}}$

because for a given angular frequency is ${\displaystyle \omega }$

${\displaystyle k_{z}^{2}<k^{2}=k_{z}^{2}+k_{T}^{2}}$

However, it shows a dispersion that leads to the following group velocity :

${\displaystyle {\begin{aligned}c_{\text{Gruppe}}&=c\cdot \left(1-{\frac {k_{T}^{2}}{2\cdot k^{2}}}\right)\\\Rightarrow c_{\text{Gruppe}}&<c\end{aligned}}}$

Wave packets in the form of a Bessel beam are therefore slower in a vacuum than the speed of light. This could even be demonstrated in 2015 on individual photons , with the effect  reaching a few ppm (millionths) of the speed of light.

This can be used for non-linear effects , the group velocity of the pump wave with the phase velocity of a red-shifted z. B. Raman-Stokes wave and thus enable an effective pumping process with short pulses in the first place.

generation

An axicon creates a Bessel-like beam

The central bright spot of the Bessel ray heals itself after a disturbance, but shifted, cf. Picture above.

Real Bessel rays can be generated just as little as an ideal plane wave , since both would require an infinite amount of energy. However, good approximations can be generated by focusing a Gaussian beam with the help of an axicon , a special, conically ground lens . The resulting Bessel Gaussian rays still have the self-healing properties of an ideal Bessel ray over a certain range.

application

Bessel rays are used for optical tweezers due to their self-healing properties .

They also offer advantages in light disk microscopy : The light disk is then created by scanning a Bessel Gaussian beam, which leads to a narrower central peak and therefore better axial resolution with the same energy introduced as a Gaussian light disk. In addition, the self-healing enables a greater depth of penetration .

literature

• Bahaa EA Saleh, Malvin Carl Teich: Fundamentals of photonics 2007, ISBN 978-0-471-35832-9

• S.Klewitz: Stimulated light scattering with Bessel rays , 1st edition Allensbach. UFO, Atelier für Gestaltung und Verl. 1998. 170 p. UFO dissertation; 335., ISBN [null 3-930803-34-8]

Individual evidence

1. ↑ J. Durnin, JH Eberly, JJ Miceli, Diffraction Free Beams , Phys. Rev. Lett., Vol. 58, 1987, pp. 1499-1501
2. ↑ ^{a b} Bahaa EA Saleh, Malvin Carl Teich: Fundamentals of photonics 2007, ISBN 978-0-471-35832-9
3. ↑ D. Giovannini et al .: Spatially structured photons that travel in free space slower than the speed of light , Science, online January 22, 2015; doi: 10.1126 / science.aaa3035
4. ↑ S. Klewitz, S. Sogomonian, M. Woerner, S. Herminghaus: Stimulated Raman scattering of femtosecond Bessel pulses. In: Optics Communications . Volume 154, Number 4, 1998, pp. 186-190, doi : 10.1016 / S0030-4018 (98) 00317-4 .
5. ↑ J. Durnin, JJ Miceli: Diffraction-free beams . In: Physical Review Letters . 58, No. 15, April 1987, pp. 1499-1501. doi : 10.1103 / PhysRevLett.58.1499 .
6. ^ F. Gori, G. Guattari, C. Padovani: Bessel-Gauss beams . In: Optics Communications . 64, No. 6, December 1987, pp. 491-495. doi : 10.1016 / 0030-4018 (87) 90276-8 .
7. ↑ J. Arlt, V. Garces-Chavez, W. Sibbett, K. Dholakia: Optical micromanipulation using a Bessel light beam . In: Optics Communications . 197, No. 4-6, October 2001, pp. 239-245. ISSN  0030-4018 . doi : 10.1016 / S0030-4018 (01) 01479-1 .
8. ^ FO Fahrbach, A. Rohrbach: A line scanned light-sheet microscope with phase-shaped self-reconstructing beams. In: Optics express. Volume 18, Number 23, November 2010, pp. 24229-24244, ISSN  1094-4087 . PMID 21164769 .
9. ^ TA Planchon, L. Gao, DE Milkie, MW Davidson, JA Galbraith, CG Galbraith, E. Betzig: Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination. In: Nature methods. Volume 8, Number 5, May 2011, pp. 417-423, ISSN  1548-7105 . doi: 10.1038 / nmeth.1586 . PMID 21378978 .