Generation and propagation of Laguerre-Gaussian beams

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The transverse structure of light is recognized as an important discovery that is widely used for encoding information onto photons [1], to enhance high-speed and long-distance communications, in quantum information technology, etc. Laguerre–Gaussian (LG) mode is a laser vortex beam with rotationally-symmetric intensity form [2]. LG beams have found a use for free-space and fiber-optic communications [3-4], sharp focusing [5], quantum optics and studies of their applicability in various fields are still ongoing. Propagation of classical LG modes is well studied nowadays [6], but relevance of LG beams encourages researches to study anomalous LG beams, like three-index LG beams.

Theoretical basics

The standard LG modes are complete set of exact and orthogonal solutions of the free-space paraxial wave equation in cylindrical coordinates . The complex amplitude of these fields in case of  or input field may be expressed as follows:

where  is the waist radius;  are integer orders;  is generalized Laguerre polynomial.

LG modes are paraxial fields with a vortex phase structure , where  is the azimuthal angle and  is azimuthal mode index, . They can also be described by second (radial) mode index . It’s known that such modes carry well-defined orbital-angular momentum (OAM) [7]. The OAM modes have been applied in both quantum and optical communication systems [8]. 

By modeling propagation of beams, we can find the intensity and the phase distribution of them. The intensity distribution is squared absolute values of the complex amplitude and the phase distribution can be determined as taking the angle of it in the complex plane. The results of the modeling can be seen in fig.1, table 1.

Fig. 1. The intensity and the phase distribution of the LG beam ():  mm, .

The standard LG modes have a vortex phase structure . But there are LG beams, that can be described with 3 indices: , and , where the third index  ¹ is associated with the different vortex phase order. The complex amplitude of these fields in the input plane () is given by

where  is the waist radius;  are integers,;  is generalized Laguerre polynomial.

Table 1

The intensity and the phase distribution of the three-index LG beam, ,  mm

In table 1 we can see the differences in the phase distribution in comparison to fig. 1. These changes are caused by adding new index, which now represents the number of times that the phase changes it value from  to  in each circle.

Results of the modeling

The propagation of LG beams can be modeled using Fourier transform and Fresnel transform. The Fourier transform allows to describe light passing through a lens. It visualizes the intensity distribution of the input field in the rear focal plane of the lens. The Fresnel transform is a paraxial diffraction integral, enables to calculate the intensity and the phase distribution of the beam at the various distances . The results are given by (fig. 2-3), (table 2-3).

Fig. 2. The intensity and the phase distribution of the LG beam after passing through a lens ():  mm.

Fig. 3. Propagation of the LG mode in free space ():  mm,  mm. The intensity and the phase distribution.

Table 2

The intensity and the phase distribution of the three-index LG beam after passing through a lens,  mm

Table 3

Propagation of the LG mode in free space,  mm. The intensity and the phase distribution.

We can see that intensity distribution of the three-index beams in the rear focal plane of the lens and after propagation in free-space has changed comparing to table 1. That results show that adding the third index for describing classical LG modes leads to changes in beam structure during propagation. So, this modification breaks modal properties, but expands the variety of generated distributions.    

Conclusion

In the present paper we have studied the standard and three-index LG beams and their basic features. We have modeled the propagation of these beams in free space and passing through a lens using Fresnel transform and Fourier transform. It has been shown that classical LG modes preserve their structure up to a scale during propagation. Three-index LG beams act differently, they lose their modal properties, but provide a greater variety of distributions.

About the authors

Марина Леонидовна Колебанова

Author for correspondence.
Email: infrocl@yandex.ru
Russian Federation

Natalia Aleksandrovna Slobozhanina

Email: slobogeanina@mail.ru

References

Supplementary files